200827

Ruby


A336940

60 = 2^2 * 3^1 * 5^1

の約数は、

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

であるが、このうち奇数のものは、

1, 3, 5, 15

である。これらは、

15 = 3^1 * 5^1

の約数である。

この考え方を使って、

n! の約数のうち奇数となるものの個数を求めてみた。

require 'prime'

def A(n)
  h = {}
  Prime.each(n * n){|i| h[i] = 1}
  ary0 = []
  ary1 = [1]
  (1..n * n).each{|i|
    i.prime_division.each{|j| h[j[0]] += j[1]}
    ary0 << h[2]
    ary1 << h.values.inject(:*) / h[2]
  }
  p ary0
  p ary1
end

A(15)

出力結果

[1, 2, 2, 4, 4, 5, 5, 8, 8, 9, 9, 11, 11, 12, 12, 16, 16, 17, 17, 19, 19, 20, 20, 23, 23, 24, 24, 26, 26, 27, 27, 32, 32, 33, 33, 35, 35, 36, 36, 39, 39, 40, 40, 42, 42, 43, 43, 47, 47, 48, 48, 50, 50, 51, 51, 54, 54, 55, 55, 57, 57, 58, 58, 64, 64, 65, 65, 67, 67, 68, 68, 71, 71, 72, 72, 74, 74, 75, 75, 79, 79, 80, 80, 82, 82, 83, 83, 86, 86, 87, 87, 89, 89, 90, 90, 95, 95, 96, 96, 98, 98, 99, 99, 102, 102, 103, 103, 105, 105, 106, 106, 110, 110, 111, 111, 113, 113, 114, 114, 117, 117, 118, 118, 120, 120, 121, 121, 128, 128, 129, 129, 131, 131, 132, 132, 135, 135, 136, 136, 138, 138, 139, 139, 143, 143, 144, 144, 146, 146, 147, 147, 150, 150, 151, 151, 153, 153, 154, 154, 159, 159, 160, 160, 162, 162, 163, 163, 166, 166, 167, 167, 169, 169, 170, 170, 174, 174, 175, 175, 177, 177, 178, 178, 181, 181, 182, 182, 184, 184, 185, 185, 191, 191, 192, 192, 194, 194, 195, 195, 198, 198, 199, 199, 201, 201, 202, 202, 206, 206, 207, 207, 209, 209, 210, 210, 213, 213, 214, 214, 216, 216, 217, 217, 222, 222]

[1, 1, 1, 2, 2, 4, 6, 12, 12, 20, 30, 60, 72, 144, 216, 336, 336, 672, 864, 1728, 2160, 3200, 4800, 9600, 10560, 14784, 22176, 28224, 35280, 70560, 86400, 172800, 172800, 245760, 368640, 497664, 559872, 1119744, 1679616, 2363904, 2626560, 5253120, 6451200, 12902400, 16128000, 19514880, 29272320, 58544640, 61205760, 78693120, 93000960, 129392640, 161740800, 323481600, 363916800, 470292480, 522547200, 722534400, 1083801600, 2167603200, 2405376000, 4810752000, 7216128000, 8485171200, 8485171200, 10861019136, 13079937024, 26159874048, 32699842560, 44962283520, 52115374080, 104230748160, 110547763200, 221095526400, 331643289600, 381250437120, 476563046400, 590030438400, 707490201600, 1414980403200, 1489453056000, 1650475008000, 2475712512000, 4951425024000, 5462360064000, 6882573680640, 10323860520960, 14092888965120, 15854500085760, 31709000171520, 34764020121600, 42568187904000, 53210234880000, 72523579392000, 108785369088000, 136476190310400, 139443064012800, 278886128025600, 316070945095680, 366134191718400, 397971947520000, 795943895040000, 947552256000000, 1895104512000000, 2131992576000000, 2394654061363200, 3591981092044800, 7183962184089600, 7606548194918400, 15213096389836800, 17378037029928960, 23599803373977600, 24910903561420800, 49821807122841600, 59182267855011840, 73649044441792512, 92061305552240640, 105943565913292800, 158915348869939200, 191176359542784000, 201417950232576000, 238039395729408000, 357059093594112000, 484147923517440000, 605184904396800000, 667790239334400000, 724552409677824000, 1449104819355648000, 1449104819355648000, 1963303303643136000, 2227122185070182400, 4454244370140364800, 4873019481862963200, 5834363461278105600, 8751545191917158400, 9439403384084889600, 10619328807095500800, 21238657614191001600, 25148261254614220800, 50296522509228441600, 54129278903648256000, 73233730281406464000, 109850595422109696000, 128396799844024320000, 132118446216314880000, 163071910758422937600, 244607866137634406400, 269622877555261440000, 337028596944076800000, 674057193888153600000, 721386827100979200000, 1442773654201958400000, 1623120360977203200000, 1852877124403200000000, 2055458356671283200000, 2531459239268843520000, 2778979698219574886400, 5557959396439149772800, 8336939094658724659200, 11262180882258277171200, 11550954751034130432000, 13708825418809737216000, 14420972193812840448000, 28841944387625680896000, 36052430484532101120000, 39748082791530627072000, 59622124187295940608000, 119244248374591881216000, 125168759359779373056000, 144425491568976199680000, 162742870987480498176000, 185182651056704716800000, 231478313820880896000000, 462956627641761792000000, 546470372235961958400000, 592938941303645798400000, 627817702556801433600000, 846823877867313561600000, 1270235816800970342400000, 2540471633601940684800000, 2657938582177579008000000, 5315877164355158016000000, 5865795491702243328000000, 7908937741621002240000000, 8897554959323627520000000, 10914334083436983091200000, 12874871872498811535360000, 14825610034998631464960000, 18532012543748289331200000, 19781056612630957916160000, 22232187540717837484032000, 44464375081435674968064000, 44937400348259458744320000, 89874800696518917488640000, 134812201044778376232960000, 147825428066122044801024000, 157362552457484757368832000, 314725104914969514737664000, 338191450456875136450560000, 676382900913750272901120000, 704565521785156534272000000, 949006621180006760448000000, 1423509931770010140672000000, 1676167538707544408064000000, 1834190067609265766400000000, 2245048642753741298073600000, 3367572964130611947110400000, 3816582692681360206725120000, 4041087556956734336532480000, 4628882110695895694573568000, 4906089515746503646248960000, 9812179031493007292497920000, 12265223789366259115622400000, 16512404842641954019737600000, 24768607263962931029606400000, 30293911961308507951595520000, 31167774806346253373276160000, 36638037323378452944912384000, 54957055985067679417368576000, 73960897774296690057019392000, 78944785764478407715848192000, 89740653903210497659895808000, 105666850197761743386574848000, 211333700395523486773149696000, 217204080962065805850181632000, 229381679792762665505390592000]

200816

PARI


PARI と OneDrive

\r ○○.gp でファイルを読み込むとき、

ファイルがOneDrive 上にあると読み込めないことに

気づいた。

Windows 10 だと、特に設定し直さないかぎりデスクトップも

OneDrive の下のフォルダになっていることが多いので注意しよう。

200811

Vue.js

select

<select v-model="プロパティ名" multiple>

の複数選択について、

Ctrl を押しながら項目を選ぶというのは不便だと思う。

200808

Ruby


A243631 とA336727

A307883 とA307884 と似た関係式をもつ。

# A307883とA307884
def A(k, n)
  i, a, b = 0, 1, k + 1
  ary = [1]
  while i < n
    i += 1
    a, b = b, ((k + 1) * (2 * i + 1) * b - (k - 1) ** 2 * i * a) / (i + 1)
    ary << a
  end
  ary
end

# A243631とA336727
def B(k, n)
  i, a, b = 0, 1, 1
  ary = [1]
  while i < n
    i += 1
    a, b = b, ((k + 1) * (2 * i + 1) * b - (k - 1) ** 2 * (i - 1) * a) / (i + 2)
    ary << a
  end
  ary
end

n = 10
-10.upto(10){|i| p [i, A(i, n)]}
-10.upto(10){|i| p [i, B(i, n)]}

出力結果

[-10, [1, -9, 61, -189, -2559, 59751, -727859, 5968611, -23658239, -239767209, 6676401501]]

[-9, [1, -8, 46, -80, -2330, 39952, -391796, 2396512, -1665530, -187855280, 3005297956]]

[-8, [1, -7, 33, -7, -1919, 24633, -186591, 715449, 3834369, -102211207, 1079883553]]

[-7, [1, -6, 22, 36, -1434, 13644, -73604, 71688, 3315334, -41652036, 283869972]]

[-6, [1, -5, 13, 55, -959, 6475, -20195, -84425, 1657345, -11975525, 40678573]]

[-5, [1, -4, 6, 56, -554, 2376, -804, -67344, 530406, -1852504, -3106124]]

[-4, [1, -3, 1, 45, -255, 477, 2689, -25203, 82945, 90045, -2379519]]

[-3, [1, -2, -2, 28, -74, -92, 1324, -3656, -4826, 70228, -197372]]

[-2, [1, -1, -3, 11, 1, -81, 141, 363, -1791, 479, 13597]]

[-1, [1, 0, -2, 0, 6, 0, -20, 0, 70, 0, -252]]

[0, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]]

[1, [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756]]

[2, [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453]]

[3, [1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544, 98365972]]

[4, [1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525, 653808673]]

[5, [1, 6, 46, 396, 3606, 33876, 324556, 3151896, 30915046, 305543556, 3038019876]]

[6, [1, 7, 61, 595, 6145, 65527, 712909, 7863667, 87615745, 983726695, 11112210781]]

[7, [1, 8, 78, 848, 9766, 116208, 1411404, 17383584, 216294534, 2712176048, 34217133028]]

[8, [1, 9, 97, 1161, 14721, 192969, 2582881, 35066313, 481003009, 6649718409, 92497952097]]

[9, [1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300, 225759595348]]

[10, [1, 11, 141, 1991, 29761, 460251, 7272861, 116619591, 1889815041, 30869546411, 507406003501]]

[-10, [1, 1, -9, 71, -459, 1901, 4851, -214709, 2829321, -25102759, 141253551]]

[-9, [1, 1, -8, 55, -296, 802, 6832, -138941, 1397080, -9274418, 26549776]]

[-8, [1, 1, -7, 41, -175, 177, 6153, -78951, 588833, -2530591, -4090471]]

[-7, [1, 1, -6, 29, -90, -118, 4404, -38219, 194286, -269506, -6250068]]

[-6, [1, 1, -5, 19, -35, -203, 2575, -14705, 38425, 177769, -2904605]]

[-5, [1, 1, -4, 11, -4, -174, 1176, -3729, -3364, 116846, -719224]]

[-4, [1, 1, -3, 5, 9, -103, 357, -131, -5295, 29297, -55539]]

[-3, [1, 1, -2, 1, 10, -38, 28, 289, -1262, 1054, 11044]]

[-2, [1, 1, -1, -1, 5, -3, -21, 51, 41, -391, 407]]

[-1, [1, 1, 0, -1, 0, 2, 0, -5, 0, 14, 0]]

[0, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]]

[1, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796]]

[2, [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859]]

[3, [1, 1, 4, 19, 100, 562, 3304, 20071, 124996, 793774, 5120632]]

[4, [1, 1, 5, 29, 185, 1257, 8925, 65445, 491825, 3768209, 29324405]]

[5, [1, 1, 6, 41, 306, 2426, 20076, 171481, 1500666, 13386206, 121267476]]

[6, [1, 1, 7, 55, 469, 4237, 39907, 387739, 3858505, 39130777, 402972031]]

[7, [1, 1, 8, 71, 680, 6882, 72528, 788019, 8766248, 99362894, 1143498224]]

[8, [1, 1, 9, 89, 945, 10577, 123129, 1476841, 18130401, 226739489, 2878666857]]

[9, [1, 1, 10, 109, 1270, 15562, 198100, 2596645, 34825150, 475697854, 6595646860]]

[10, [1, 1, 11, 131, 1661, 22101, 305151, 4335711, 63009881, 932449961, 14004694451]]

200725

Ruby


Martin Gardner のlucky number について

2187 + 1234 = 3421
2187 + 12345 = 14532
2187 + 123456 = 125643
2187 + 1234567 = 1236754
2187 + 12345678 = 12347865
2187 + 123456789 = 123458976

のうち、

2187 + 1234 = 3421
2187 + 2345 = 4532
2187 + 3456 = 5643
2187 + 4567 = 6754
2187 + 5678 = 7865
2187 + 6789 = 8976

の部分が大事だと思うので、このような数字を見つけてみた。

def A(digit)
  ary = []
  (10 ** (digit - 1)..10 ** digit - 1).each{|i|
    flag = true
    (1..10 - digit).each{|j|
      a = (j..j + digit - 1).to_a
      k = a.join.to_i
      l = i + k
      if l.to_s.split('').map(&:to_i).sort != a
        flag = false
        break
      end
    }
    ary << i if flag
  }
  ary
end

def B(ary, digit)
  p ary
  puts
  ary.each{|i|
    (1..10 - digit).each{|j|
      k = (j..j + digit - 1).to_a.join.to_i
      puts "#{i} + #{k} = #{i + k}"
    }
    puts "-" * (digit * 3 + 6)
  }
end

(1..5).each{|i| B(A(i), i)}

出力結果

[]

[]

[108, 189, 198]

108 + 123 = 231

108 + 234 = 342

108 + 345 = 453

108 + 456 = 564

108 + 567 = 675

108 + 678 = 786

108 + 789 = 897

---------------

189 + 123 = 312

189 + 234 = 423

189 + 345 = 534

189 + 456 = 645

189 + 567 = 756

189 + 678 = 867

189 + 789 = 978

---------------

198 + 123 = 321

198 + 234 = 432

198 + 345 = 543

198 + 456 = 654

198 + 567 = 765

198 + 678 = 876

198 + 789 = 987

---------------

[1080, 1107, 1179, 1197, 1890, 1908, 1980, 2007, 2178, 2187, 2889, 2898, 2979, 2997, 3078, 3087]

1080 + 1234 = 2314

1080 + 2345 = 3425

1080 + 3456 = 4536

1080 + 4567 = 5647

1080 + 5678 = 6758

1080 + 6789 = 7869

------------------

1107 + 1234 = 2341

1107 + 2345 = 3452

1107 + 3456 = 4563

1107 + 4567 = 5674

1107 + 5678 = 6785

1107 + 6789 = 7896

------------------

1179 + 1234 = 2413

1179 + 2345 = 3524

1179 + 3456 = 4635

1179 + 4567 = 5746

1179 + 5678 = 6857

1179 + 6789 = 7968

------------------

1197 + 1234 = 2431

1197 + 2345 = 3542

1197 + 3456 = 4653

1197 + 4567 = 5764

1197 + 5678 = 6875

1197 + 6789 = 7986

------------------

1890 + 1234 = 3124

1890 + 2345 = 4235

1890 + 3456 = 5346

1890 + 4567 = 6457

1890 + 5678 = 7568

1890 + 6789 = 8679

------------------

1908 + 1234 = 3142

1908 + 2345 = 4253

1908 + 3456 = 5364

1908 + 4567 = 6475

1908 + 5678 = 7586

1908 + 6789 = 8697

------------------

1980 + 1234 = 3214

1980 + 2345 = 4325

1980 + 3456 = 5436

1980 + 4567 = 6547

1980 + 5678 = 7658

1980 + 6789 = 8769

------------------

2007 + 1234 = 3241

2007 + 2345 = 4352

2007 + 3456 = 5463

2007 + 4567 = 6574

2007 + 5678 = 7685

2007 + 6789 = 8796

------------------

2178 + 1234 = 3412

2178 + 2345 = 4523

2178 + 3456 = 5634

2178 + 4567 = 6745

2178 + 5678 = 7856

2178 + 6789 = 8967

------------------

2187 + 1234 = 3421

2187 + 2345 = 4532

2187 + 3456 = 5643

2187 + 4567 = 6754

2187 + 5678 = 7865

2187 + 6789 = 8976

------------------

2889 + 1234 = 4123

2889 + 2345 = 5234

2889 + 3456 = 6345

2889 + 4567 = 7456

2889 + 5678 = 8567

2889 + 6789 = 9678

------------------

2898 + 1234 = 4132

2898 + 2345 = 5243

2898 + 3456 = 6354

2898 + 4567 = 7465

2898 + 5678 = 8576

2898 + 6789 = 9687

------------------

2979 + 1234 = 4213

2979 + 2345 = 5324

2979 + 3456 = 6435

2979 + 4567 = 7546

2979 + 5678 = 8657

2979 + 6789 = 9768

------------------

2997 + 1234 = 4231

2997 + 2345 = 5342

2997 + 3456 = 6453

2997 + 4567 = 7564

2997 + 5678 = 8675

2997 + 6789 = 9786

------------------

3078 + 1234 = 4312

3078 + 2345 = 5423

3078 + 3456 = 6534

3078 + 4567 = 7645

3078 + 5678 = 8756

3078 + 6789 = 9867

------------------

3087 + 1234 = 4321

3087 + 2345 = 5432

3087 + 3456 = 6543

3087 + 4567 = 7654

3087 + 5678 = 8765

3087 + 6789 = 9876

------------------

[10800, 10809, 11070, 11106, 11169, 11196, 11790, 11808, 11970, 12006, 12168, 12186, 12789, 12798, 12969, 12996, 13068, 13086, 18900, 18909, 19080, 19107, 19179, 19197, 19800, 19809, 20070, 20106, 20169, 20196, 21780, 21807, 21870, 21906, 22167, 22176, 22779, 22797, 22869, 22896, 23067, 23076, 28890, 28908, 28980, 29007, 29178, 29187, 29790, 29808, 29970, 30006, 30168, 30186, 30780, 30807, 30870, 30906, 31167, 31176, 32778, 32787, 32868, 32886, 32967, 32976, 38889, 38898, 38979, 38997, 39078, 39087, 39789, 39798, 39969, 39996, 40068, 40086, 40779, 40797, 40869, 40896, 41067, 41076, 41778, 41787, 41868, 41886, 41967, 41976]

10800 + 12345 = 23145

10800 + 23456 = 34256

10800 + 34567 = 45367

10800 + 45678 = 56478

10800 + 56789 = 67589

---------------------

10809 + 12345 = 23154

10809 + 23456 = 34265

10809 + 34567 = 45376

10809 + 45678 = 56487

10809 + 56789 = 67598

---------------------

11070 + 12345 = 23415

11070 + 23456 = 34526

11070 + 34567 = 45637

11070 + 45678 = 56748

11070 + 56789 = 67859

---------------------

11106 + 12345 = 23451

11106 + 23456 = 34562

11106 + 34567 = 45673

11106 + 45678 = 56784

11106 + 56789 = 67895

---------------------

11169 + 12345 = 23514

11169 + 23456 = 34625

11169 + 34567 = 45736

11169 + 45678 = 56847

11169 + 56789 = 67958

---------------------

11196 + 12345 = 23541

11196 + 23456 = 34652

11196 + 34567 = 45763

11196 + 45678 = 56874

11196 + 56789 = 67985

---------------------

11790 + 12345 = 24135

11790 + 23456 = 35246

11790 + 34567 = 46357

11790 + 45678 = 57468

11790 + 56789 = 68579

---------------------

11808 + 12345 = 24153

11808 + 23456 = 35264

11808 + 34567 = 46375

11808 + 45678 = 57486

11808 + 56789 = 68597

---------------------

11970 + 12345 = 24315

11970 + 23456 = 35426

11970 + 34567 = 46537

11970 + 45678 = 57648

11970 + 56789 = 68759

---------------------

12006 + 12345 = 24351

12006 + 23456 = 35462

12006 + 34567 = 46573

12006 + 45678 = 57684

12006 + 56789 = 68795

---------------------

12168 + 12345 = 24513

12168 + 23456 = 35624

12168 + 34567 = 46735

12168 + 45678 = 57846

12168 + 56789 = 68957

---------------------

12186 + 12345 = 24531

12186 + 23456 = 35642

12186 + 34567 = 46753

12186 + 45678 = 57864

12186 + 56789 = 68975

---------------------

12789 + 12345 = 25134

12789 + 23456 = 36245

12789 + 34567 = 47356

12789 + 45678 = 58467

12789 + 56789 = 69578

---------------------

12798 + 12345 = 25143

12798 + 23456 = 36254

12798 + 34567 = 47365

12798 + 45678 = 58476

12798 + 56789 = 69587

---------------------

12969 + 12345 = 25314

12969 + 23456 = 36425

12969 + 34567 = 47536

12969 + 45678 = 58647

12969 + 56789 = 69758

---------------------

12996 + 12345 = 25341

12996 + 23456 = 36452

12996 + 34567 = 47563

12996 + 45678 = 58674

12996 + 56789 = 69785

---------------------

13068 + 12345 = 25413

13068 + 23456 = 36524

13068 + 34567 = 47635

13068 + 45678 = 58746

13068 + 56789 = 69857

---------------------

13086 + 12345 = 25431

13086 + 23456 = 36542

13086 + 34567 = 47653

13086 + 45678 = 58764

13086 + 56789 = 69875

---------------------

18900 + 12345 = 31245

18900 + 23456 = 42356

18900 + 34567 = 53467

18900 + 45678 = 64578

18900 + 56789 = 75689

---------------------

18909 + 12345 = 31254

18909 + 23456 = 42365

18909 + 34567 = 53476

18909 + 45678 = 64587

18909 + 56789 = 75698

---------------------

19080 + 12345 = 31425

19080 + 23456 = 42536

19080 + 34567 = 53647

19080 + 45678 = 64758

19080 + 56789 = 75869

---------------------

19107 + 12345 = 31452

19107 + 23456 = 42563

19107 + 34567 = 53674

19107 + 45678 = 64785

19107 + 56789 = 75896

---------------------

19179 + 12345 = 31524

19179 + 23456 = 42635

19179 + 34567 = 53746

19179 + 45678 = 64857

19179 + 56789 = 75968

---------------------

19197 + 12345 = 31542

19197 + 23456 = 42653

19197 + 34567 = 53764

19197 + 45678 = 64875

19197 + 56789 = 75986

---------------------

19800 + 12345 = 32145

19800 + 23456 = 43256

19800 + 34567 = 54367

19800 + 45678 = 65478

19800 + 56789 = 76589

---------------------

19809 + 12345 = 32154

19809 + 23456 = 43265

19809 + 34567 = 54376

19809 + 45678 = 65487

19809 + 56789 = 76598

---------------------

20070 + 12345 = 32415

20070 + 23456 = 43526

20070 + 34567 = 54637

20070 + 45678 = 65748

20070 + 56789 = 76859

---------------------

20106 + 12345 = 32451

20106 + 23456 = 43562

20106 + 34567 = 54673

20106 + 45678 = 65784

20106 + 56789 = 76895

---------------------

20169 + 12345 = 32514

20169 + 23456 = 43625

20169 + 34567 = 54736

20169 + 45678 = 65847

20169 + 56789 = 76958

---------------------

20196 + 12345 = 32541

20196 + 23456 = 43652

20196 + 34567 = 54763

20196 + 45678 = 65874

20196 + 56789 = 76985

---------------------

21780 + 12345 = 34125

21780 + 23456 = 45236

21780 + 34567 = 56347

21780 + 45678 = 67458

21780 + 56789 = 78569

---------------------

21807 + 12345 = 34152

21807 + 23456 = 45263

21807 + 34567 = 56374

21807 + 45678 = 67485

21807 + 56789 = 78596

---------------------

21870 + 12345 = 34215

21870 + 23456 = 45326

21870 + 34567 = 56437

21870 + 45678 = 67548

21870 + 56789 = 78659

---------------------

21906 + 12345 = 34251

21906 + 23456 = 45362

21906 + 34567 = 56473

21906 + 45678 = 67584

21906 + 56789 = 78695

---------------------

22167 + 12345 = 34512

22167 + 23456 = 45623

22167 + 34567 = 56734

22167 + 45678 = 67845

22167 + 56789 = 78956

---------------------

22176 + 12345 = 34521

22176 + 23456 = 45632

22176 + 34567 = 56743

22176 + 45678 = 67854

22176 + 56789 = 78965

---------------------

22779 + 12345 = 35124

22779 + 23456 = 46235

22779 + 34567 = 57346

22779 + 45678 = 68457

22779 + 56789 = 79568

---------------------

22797 + 12345 = 35142

22797 + 23456 = 46253

22797 + 34567 = 57364

22797 + 45678 = 68475

22797 + 56789 = 79586

---------------------

22869 + 12345 = 35214

22869 + 23456 = 46325

22869 + 34567 = 57436

22869 + 45678 = 68547

22869 + 56789 = 79658

---------------------

22896 + 12345 = 35241

22896 + 23456 = 46352

22896 + 34567 = 57463

22896 + 45678 = 68574

22896 + 56789 = 79685

---------------------

23067 + 12345 = 35412

23067 + 23456 = 46523

23067 + 34567 = 57634

23067 + 45678 = 68745

23067 + 56789 = 79856

---------------------

23076 + 12345 = 35421

23076 + 23456 = 46532

23076 + 34567 = 57643

23076 + 45678 = 68754

23076 + 56789 = 79865

---------------------

28890 + 12345 = 41235

28890 + 23456 = 52346

28890 + 34567 = 63457

28890 + 45678 = 74568

28890 + 56789 = 85679

---------------------

28908 + 12345 = 41253

28908 + 23456 = 52364

28908 + 34567 = 63475

28908 + 45678 = 74586

28908 + 56789 = 85697

---------------------

28980 + 12345 = 41325

28980 + 23456 = 52436

28980 + 34567 = 63547

28980 + 45678 = 74658

28980 + 56789 = 85769

---------------------

29007 + 12345 = 41352

29007 + 23456 = 52463

29007 + 34567 = 63574

29007 + 45678 = 74685

29007 + 56789 = 85796

---------------------

29178 + 12345 = 41523

29178 + 23456 = 52634

29178 + 34567 = 63745

29178 + 45678 = 74856

29178 + 56789 = 85967

---------------------

29187 + 12345 = 41532

29187 + 23456 = 52643

29187 + 34567 = 63754

29187 + 45678 = 74865

29187 + 56789 = 85976

---------------------

29790 + 12345 = 42135

29790 + 23456 = 53246

29790 + 34567 = 64357

29790 + 45678 = 75468

29790 + 56789 = 86579

---------------------

29808 + 12345 = 42153

29808 + 23456 = 53264

29808 + 34567 = 64375

29808 + 45678 = 75486

29808 + 56789 = 86597

---------------------

29970 + 12345 = 42315

29970 + 23456 = 53426

29970 + 34567 = 64537

29970 + 45678 = 75648

29970 + 56789 = 86759

---------------------

30006 + 12345 = 42351

30006 + 23456 = 53462

30006 + 34567 = 64573

30006 + 45678 = 75684

30006 + 56789 = 86795

---------------------

30168 + 12345 = 42513

30168 + 23456 = 53624

30168 + 34567 = 64735

30168 + 45678 = 75846

30168 + 56789 = 86957

---------------------

30186 + 12345 = 42531

30186 + 23456 = 53642

30186 + 34567 = 64753

30186 + 45678 = 75864

30186 + 56789 = 86975

---------------------

30780 + 12345 = 43125

30780 + 23456 = 54236

30780 + 34567 = 65347

30780 + 45678 = 76458

30780 + 56789 = 87569

---------------------

30807 + 12345 = 43152

30807 + 23456 = 54263

30807 + 34567 = 65374

30807 + 45678 = 76485

30807 + 56789 = 87596

---------------------

30870 + 12345 = 43215

30870 + 23456 = 54326

30870 + 34567 = 65437

30870 + 45678 = 76548

30870 + 56789 = 87659

---------------------

30906 + 12345 = 43251

30906 + 23456 = 54362

30906 + 34567 = 65473

30906 + 45678 = 76584

30906 + 56789 = 87695

---------------------

31167 + 12345 = 43512

31167 + 23456 = 54623

31167 + 34567 = 65734

31167 + 45678 = 76845

31167 + 56789 = 87956

---------------------

31176 + 12345 = 43521

31176 + 23456 = 54632

31176 + 34567 = 65743

31176 + 45678 = 76854

31176 + 56789 = 87965

---------------------

32778 + 12345 = 45123

32778 + 23456 = 56234

32778 + 34567 = 67345

32778 + 45678 = 78456

32778 + 56789 = 89567

---------------------

32787 + 12345 = 45132

32787 + 23456 = 56243

32787 + 34567 = 67354

32787 + 45678 = 78465

32787 + 56789 = 89576

---------------------

32868 + 12345 = 45213

32868 + 23456 = 56324

32868 + 34567 = 67435

32868 + 45678 = 78546

32868 + 56789 = 89657

---------------------

32886 + 12345 = 45231

32886 + 23456 = 56342

32886 + 34567 = 67453

32886 + 45678 = 78564

32886 + 56789 = 89675

---------------------

32967 + 12345 = 45312

32967 + 23456 = 56423

32967 + 34567 = 67534

32967 + 45678 = 78645

32967 + 56789 = 89756

---------------------

32976 + 12345 = 45321

32976 + 23456 = 56432

32976 + 34567 = 67543

32976 + 45678 = 78654

32976 + 56789 = 89765

---------------------

38889 + 12345 = 51234

38889 + 23456 = 62345

38889 + 34567 = 73456

38889 + 45678 = 84567

38889 + 56789 = 95678

---------------------

38898 + 12345 = 51243

38898 + 23456 = 62354

38898 + 34567 = 73465

38898 + 45678 = 84576

38898 + 56789 = 95687

---------------------

38979 + 12345 = 51324

38979 + 23456 = 62435

38979 + 34567 = 73546

38979 + 45678 = 84657

38979 + 56789 = 95768

---------------------

38997 + 12345 = 51342

38997 + 23456 = 62453

38997 + 34567 = 73564

38997 + 45678 = 84675

38997 + 56789 = 95786

---------------------

39078 + 12345 = 51423

39078 + 23456 = 62534

39078 + 34567 = 73645

39078 + 45678 = 84756

39078 + 56789 = 95867

---------------------

39087 + 12345 = 51432

39087 + 23456 = 62543

39087 + 34567 = 73654

39087 + 45678 = 84765

39087 + 56789 = 95876

---------------------

39789 + 12345 = 52134

39789 + 23456 = 63245

39789 + 34567 = 74356

39789 + 45678 = 85467

39789 + 56789 = 96578

---------------------

39798 + 12345 = 52143

39798 + 23456 = 63254

39798 + 34567 = 74365

39798 + 45678 = 85476

39798 + 56789 = 96587

---------------------

39969 + 12345 = 52314

39969 + 23456 = 63425

39969 + 34567 = 74536

39969 + 45678 = 85647

39969 + 56789 = 96758

---------------------

39996 + 12345 = 52341

39996 + 23456 = 63452

39996 + 34567 = 74563

39996 + 45678 = 85674

39996 + 56789 = 96785

---------------------

40068 + 12345 = 52413

40068 + 23456 = 63524

40068 + 34567 = 74635

40068 + 45678 = 85746

40068 + 56789 = 96857

---------------------

40086 + 12345 = 52431

40086 + 23456 = 63542

40086 + 34567 = 74653

40086 + 45678 = 85764

40086 + 56789 = 96875

---------------------

40779 + 12345 = 53124

40779 + 23456 = 64235

40779 + 34567 = 75346

40779 + 45678 = 86457

40779 + 56789 = 97568

---------------------

40797 + 12345 = 53142

40797 + 23456 = 64253

40797 + 34567 = 75364

40797 + 45678 = 86475

40797 + 56789 = 97586

---------------------

40869 + 12345 = 53214

40869 + 23456 = 64325

40869 + 34567 = 75436

40869 + 45678 = 86547

40869 + 56789 = 97658

---------------------

40896 + 12345 = 53241

40896 + 23456 = 64352

40896 + 34567 = 75463

40896 + 45678 = 86574

40896 + 56789 = 97685

---------------------

41067 + 12345 = 53412

41067 + 23456 = 64523

41067 + 34567 = 75634

41067 + 45678 = 86745

41067 + 56789 = 97856

---------------------

41076 + 12345 = 53421

41076 + 23456 = 64532

41076 + 34567 = 75643

41076 + 45678 = 86754

41076 + 56789 = 97865

---------------------

41778 + 12345 = 54123

41778 + 23456 = 65234

41778 + 34567 = 76345

41778 + 45678 = 87456

41778 + 56789 = 98567

---------------------

41787 + 12345 = 54132

41787 + 23456 = 65243

41787 + 34567 = 76354

41787 + 45678 = 87465

41787 + 56789 = 98576

---------------------

41868 + 12345 = 54213

41868 + 23456 = 65324

41868 + 34567 = 76435

41868 + 45678 = 87546

41868 + 56789 = 98657

---------------------

41886 + 12345 = 54231

41886 + 23456 = 65342

41886 + 34567 = 76453

41886 + 45678 = 87564

41886 + 56789 = 98675

---------------------

41967 + 12345 = 54312

41967 + 23456 = 65423

41967 + 34567 = 76534

41967 + 45678 = 87645

41967 + 56789 = 98756

---------------------

41976 + 12345 = 54321

41976 + 23456 = 65432

41976 + 34567 = 76543

41976 + 45678 = 87654

41976 + 56789 = 98765

---------------------

200723

PARI


A307883 とA307884

diagonal of the rational function R(r, k) = r / ((1-r*x)*(1-r*y) + r-1 - (k+r-1)*r*x*y)

がr (≠0) の値に依存しないことは、以下のように展開すれば確認できる。

R(r, k)

= r / ((1-r*x-r*y+r^2*x*y) + r-1 - (k+r-1)*r*x*y)

= r / (r * (1-x-y-(k-1)*x*y))

= 1 / ((1-x)*(1-y) - k*x*y)

= R(1, k)

実際計算でも確かめてみた。

(00:00) gp > N=10;
(00:00) gp > diag(n, expr, var=variables(expr)) = {
  my(a=vector(n));
  for(i=1, #var, expr=taylor(expr, var[#var-i+1], n));
  for(j=1, n, a[j]=expr;
    for(i=1, #var, a[j]=polcoeff(a[j], j-1)));
  return(a);
};
(00:00) gp > R(r, k) = r/((1-r*x)*(1-r*y)+r-1-(k+r-1)*r*x*y);
(00:00) gp > for(k=-10, 10, for(r=-5, 5, if(r!=0, print("(r,k) = (", r, ",", k, ") ", diag(N, R(r, k))))))
(r,k) = (-5,-10) [1, -9, 61, -189, -2559, 59751, -727859, 5968611, -23658239, -239767209]
(r,k) = (-4,-10) [1, -9, 61, -189, -2559, 59751, -727859, 5968611, -23658239, -239767209]
(r,k) = (-3,-10) [1, -9, 61, -189, -2559, 59751, -727859, 5968611, -23658239, -239767209]
(r,k) = (-2,-10) [1, -9, 61, -189, -2559, 59751, -727859, 5968611, -23658239, -239767209]
(r,k) = (-1,-10) [1, -9, 61, -189, -2559, 59751, -727859, 5968611, -23658239, -239767209]
(r,k) = (1,-10) [1, -9, 61, -189, -2559, 59751, -727859, 5968611, -23658239, -239767209]
(r,k) = (2,-10) [1, -9, 61, -189, -2559, 59751, -727859, 5968611, -23658239, -239767209]
(r,k) = (3,-10) [1, -9, 61, -189, -2559, 59751, -727859, 5968611, -23658239, -239767209]
(r,k) = (4,-10) [1, -9, 61, -189, -2559, 59751, -727859, 5968611, -23658239, -239767209]
(r,k) = (5,-10) [1, -9, 61, -189, -2559, 59751, -727859, 5968611, -23658239, -239767209]
(r,k) = (-5,-9) [1, -8, 46, -80, -2330, 39952, -391796, 2396512, -1665530, -187855280]
(r,k) = (-4,-9) [1, -8, 46, -80, -2330, 39952, -391796, 2396512, -1665530, -187855280]
(r,k) = (-3,-9) [1, -8, 46, -80, -2330, 39952, -391796, 2396512, -1665530, -187855280]
(r,k) = (-2,-9) [1, -8, 46, -80, -2330, 39952, -391796, 2396512, -1665530, -187855280]
(r,k) = (-1,-9) [1, -8, 46, -80, -2330, 39952, -391796, 2396512, -1665530, -187855280]
(r,k) = (1,-9) [1, -8, 46, -80, -2330, 39952, -391796, 2396512, -1665530, -187855280]
(r,k) = (2,-9) [1, -8, 46, -80, -2330, 39952, -391796, 2396512, -1665530, -187855280]
(r,k) = (3,-9) [1, -8, 46, -80, -2330, 39952, -391796, 2396512, -1665530, -187855280]
(r,k) = (4,-9) [1, -8, 46, -80, -2330, 39952, -391796, 2396512, -1665530, -187855280]
(r,k) = (5,-9) [1, -8, 46, -80, -2330, 39952, -391796, 2396512, -1665530, -187855280]
(r,k) = (-5,-8) [1, -7, 33, -7, -1919, 24633, -186591, 715449, 3834369, -102211207]
(r,k) = (-4,-8) [1, -7, 33, -7, -1919, 24633, -186591, 715449, 3834369, -102211207]
(r,k) = (-3,-8) [1, -7, 33, -7, -1919, 24633, -186591, 715449, 3834369, -102211207]
(r,k) = (-2,-8) [1, -7, 33, -7, -1919, 24633, -186591, 715449, 3834369, -102211207]
(r,k) = (-1,-8) [1, -7, 33, -7, -1919, 24633, -186591, 715449, 3834369, -102211207]
(r,k) = (1,-8) [1, -7, 33, -7, -1919, 24633, -186591, 715449, 3834369, -102211207]
(r,k) = (2,-8) [1, -7, 33, -7, -1919, 24633, -186591, 715449, 3834369, -102211207]
(r,k) = (3,-8) [1, -7, 33, -7, -1919, 24633, -186591, 715449, 3834369, -102211207]
(r,k) = (4,-8) [1, -7, 33, -7, -1919, 24633, -186591, 715449, 3834369, -102211207]
(r,k) = (5,-8) [1, -7, 33, -7, -1919, 24633, -186591, 715449, 3834369, -102211207]
(r,k) = (-5,-7) [1, -6, 22, 36, -1434, 13644, -73604, 71688, 3315334, -41652036]
(r,k) = (-4,-7) [1, -6, 22, 36, -1434, 13644, -73604, 71688, 3315334, -41652036]
(r,k) = (-3,-7) [1, -6, 22, 36, -1434, 13644, -73604, 71688, 3315334, -41652036]
(r,k) = (-2,-7) [1, -6, 22, 36, -1434, 13644, -73604, 71688, 3315334, -41652036]
(r,k) = (-1,-7) [1, -6, 22, 36, -1434, 13644, -73604, 71688, 3315334, -41652036]
(r,k) = (1,-7) [1, -6, 22, 36, -1434, 13644, -73604, 71688, 3315334, -41652036]
(r,k) = (2,-7) [1, -6, 22, 36, -1434, 13644, -73604, 71688, 3315334, -41652036]
(r,k) = (3,-7) [1, -6, 22, 36, -1434, 13644, -73604, 71688, 3315334, -41652036]
(r,k) = (4,-7) [1, -6, 22, 36, -1434, 13644, -73604, 71688, 3315334, -41652036]
(r,k) = (5,-7) [1, -6, 22, 36, -1434, 13644, -73604, 71688, 3315334, -41652036]
(r,k) = (-5,-6) [1, -5, 13, 55, -959, 6475, -20195, -84425, 1657345, -11975525]
(r,k) = (-4,-6) [1, -5, 13, 55, -959, 6475, -20195, -84425, 1657345, -11975525]
(r,k) = (-3,-6) [1, -5, 13, 55, -959, 6475, -20195, -84425, 1657345, -11975525]
(r,k) = (-2,-6) [1, -5, 13, 55, -959, 6475, -20195, -84425, 1657345, -11975525]
(r,k) = (-1,-6) [1, -5, 13, 55, -959, 6475, -20195, -84425, 1657345, -11975525]
(r,k) = (1,-6) [1, -5, 13, 55, -959, 6475, -20195, -84425, 1657345, -11975525]
(r,k) = (2,-6) [1, -5, 13, 55, -959, 6475, -20195, -84425, 1657345, -11975525]
(r,k) = (3,-6) [1, -5, 13, 55, -959, 6475, -20195, -84425, 1657345, -11975525]
(r,k) = (4,-6) [1, -5, 13, 55, -959, 6475, -20195, -84425, 1657345, -11975525]
(r,k) = (5,-6) [1, -5, 13, 55, -959, 6475, -20195, -84425, 1657345, -11975525]
(r,k) = (-5,-5) [1, -4, 6, 56, -554, 2376, -804, -67344, 530406, -1852504]
(r,k) = (-4,-5) [1, -4, 6, 56, -554, 2376, -804, -67344, 530406, -1852504]
(r,k) = (-3,-5) [1, -4, 6, 56, -554, 2376, -804, -67344, 530406, -1852504]
(r,k) = (-2,-5) [1, -4, 6, 56, -554, 2376, -804, -67344, 530406, -1852504]
(r,k) = (-1,-5) [1, -4, 6, 56, -554, 2376, -804, -67344, 530406, -1852504]
(r,k) = (1,-5) [1, -4, 6, 56, -554, 2376, -804, -67344, 530406, -1852504]
(r,k) = (2,-5) [1, -4, 6, 56, -554, 2376, -804, -67344, 530406, -1852504]
(r,k) = (3,-5) [1, -4, 6, 56, -554, 2376, -804, -67344, 530406, -1852504]
(r,k) = (4,-5) [1, -4, 6, 56, -554, 2376, -804, -67344, 530406, -1852504]
(r,k) = (5,-5) [1, -4, 6, 56, -554, 2376, -804, -67344, 530406, -1852504]
(r,k) = (-5,-4) [1, -3, 1, 45, -255, 477, 2689, -25203, 82945, 90045]
(r,k) = (-4,-4) [1, -3, 1, 45, -255, 477, 2689, -25203, 82945, 90045]
(r,k) = (-3,-4) [1, -3, 1, 45, -255, 477, 2689, -25203, 82945, 90045]
(r,k) = (-2,-4) [1, -3, 1, 45, -255, 477, 2689, -25203, 82945, 90045]
(r,k) = (-1,-4) [1, -3, 1, 45, -255, 477, 2689, -25203, 82945, 90045]
(r,k) = (1,-4) [1, -3, 1, 45, -255, 477, 2689, -25203, 82945, 90045]
(r,k) = (2,-4) [1, -3, 1, 45, -255, 477, 2689, -25203, 82945, 90045]
(r,k) = (3,-4) [1, -3, 1, 45, -255, 477, 2689, -25203, 82945, 90045]
(r,k) = (4,-4) [1, -3, 1, 45, -255, 477, 2689, -25203, 82945, 90045]
(r,k) = (5,-4) [1, -3, 1, 45, -255, 477, 2689, -25203, 82945, 90045]
(r,k) = (-5,-3) [1, -2, -2, 28, -74, -92, 1324, -3656, -4826, 70228]
(r,k) = (-4,-3) [1, -2, -2, 28, -74, -92, 1324, -3656, -4826, 70228]
(r,k) = (-3,-3) [1, -2, -2, 28, -74, -92, 1324, -3656, -4826, 70228]
(r,k) = (-2,-3) [1, -2, -2, 28, -74, -92, 1324, -3656, -4826, 70228]
(r,k) = (-1,-3) [1, -2, -2, 28, -74, -92, 1324, -3656, -4826, 70228]
(r,k) = (1,-3) [1, -2, -2, 28, -74, -92, 1324, -3656, -4826, 70228]
(r,k) = (2,-3) [1, -2, -2, 28, -74, -92, 1324, -3656, -4826, 70228]
(r,k) = (3,-3) [1, -2, -2, 28, -74, -92, 1324, -3656, -4826, 70228]
(r,k) = (4,-3) [1, -2, -2, 28, -74, -92, 1324, -3656, -4826, 70228]
(r,k) = (5,-3) [1, -2, -2, 28, -74, -92, 1324, -3656, -4826, 70228]
(r,k) = (-5,-2) [1, -1, -3, 11, 1, -81, 141, 363, -1791, 479]
(r,k) = (-4,-2) [1, -1, -3, 11, 1, -81, 141, 363, -1791, 479]
(r,k) = (-3,-2) [1, -1, -3, 11, 1, -81, 141, 363, -1791, 479]
(r,k) = (-2,-2) [1, -1, -3, 11, 1, -81, 141, 363, -1791, 479]
(r,k) = (-1,-2) [1, -1, -3, 11, 1, -81, 141, 363, -1791, 479]
(r,k) = (1,-2) [1, -1, -3, 11, 1, -81, 141, 363, -1791, 479]
(r,k) = (2,-2) [1, -1, -3, 11, 1, -81, 141, 363, -1791, 479]
(r,k) = (3,-2) [1, -1, -3, 11, 1, -81, 141, 363, -1791, 479]
(r,k) = (4,-2) [1, -1, -3, 11, 1, -81, 141, 363, -1791, 479]
(r,k) = (5,-2) [1, -1, -3, 11, 1, -81, 141, 363, -1791, 479]
(r,k) = (-5,-1) [1, 0, -2, 0, 6, 0, -20, 0, 70, 0]
(r,k) = (-4,-1) [1, 0, -2, 0, 6, 0, -20, 0, 70, 0]
(r,k) = (-3,-1) [1, 0, -2, 0, 6, 0, -20, 0, 70, 0]
(r,k) = (-2,-1) [1, 0, -2, 0, 6, 0, -20, 0, 70, 0]
(r,k) = (-1,-1) [1, 0, -2, 0, 6, 0, -20, 0, 70, 0]
(r,k) = (1,-1) [1, 0, -2, 0, 6, 0, -20, 0, 70, 0]
(r,k) = (2,-1) [1, 0, -2, 0, 6, 0, -20, 0, 70, 0]
(r,k) = (3,-1) [1, 0, -2, 0, 6, 0, -20, 0, 70, 0]
(r,k) = (4,-1) [1, 0, -2, 0, 6, 0, -20, 0, 70, 0]
(r,k) = (5,-1) [1, 0, -2, 0, 6, 0, -20, 0, 70, 0]
(r,k) = (-5,0) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(r,k) = (-4,0) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(r,k) = (-3,0) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(r,k) = (-2,0) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(r,k) = (-1,0) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(r,k) = (1,0) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(r,k) = (2,0) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(r,k) = (3,0) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(r,k) = (4,0) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(r,k) = (5,0) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(r,k) = (-5,1) [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
(r,k) = (-4,1) [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
(r,k) = (-3,1) [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
(r,k) = (-2,1) [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
(r,k) = (-1,1) [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
(r,k) = (1,1) [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
(r,k) = (2,1) [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
(r,k) = (3,1) [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
(r,k) = (4,1) [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
(r,k) = (5,1) [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
(r,k) = (-5,2) [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563]
(r,k) = (-4,2) [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563]
(r,k) = (-3,2) [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563]
(r,k) = (-2,2) [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563]
(r,k) = (-1,2) [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563]
(r,k) = (1,2) [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563]
(r,k) = (2,2) [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563]
(r,k) = (3,2) [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563]
(r,k) = (4,2) [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563]
(r,k) = (5,2) [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563]
(r,k) = (-5,3) [1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544]
(r,k) = (-4,3) [1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544]
(r,k) = (-3,3) [1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544]
(r,k) = (-2,3) [1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544]
(r,k) = (-1,3) [1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544]
(r,k) = (1,3) [1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544]
(r,k) = (2,3) [1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544]
(r,k) = (3,3) [1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544]
(r,k) = (4,3) [1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544]
(r,k) = (5,3) [1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544]
(r,k) = (-5,4) [1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525]
(r,k) = (-4,4) [1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525]
(r,k) = (-3,4) [1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525]
(r,k) = (-2,4) [1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525]
(r,k) = (-1,4) [1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525]
(r,k) = (1,4) [1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525]
(r,k) = (2,4) [1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525]
(r,k) = (3,4) [1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525]
(r,k) = (4,4) [1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525]
(r,k) = (5,4) [1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525]
(r,k) = (-5,5) [1, 6, 46, 396, 3606, 33876, 324556, 3151896, 30915046, 305543556]
(r,k) = (-4,5) [1, 6, 46, 396, 3606, 33876, 324556, 3151896, 30915046, 305543556]
(r,k) = (-3,5) [1, 6, 46, 396, 3606, 33876, 324556, 3151896, 30915046, 305543556]
(r,k) = (-2,5) [1, 6, 46, 396, 3606, 33876, 324556, 3151896, 30915046, 305543556]
(r,k) = (-1,5) [1, 6, 46, 396, 3606, 33876, 324556, 3151896, 30915046, 305543556]
(r,k) = (1,5) [1, 6, 46, 396, 3606, 33876, 324556, 3151896, 30915046, 305543556]
(r,k) = (2,5) [1, 6, 46, 396, 3606, 33876, 324556, 3151896, 30915046, 305543556]
(r,k) = (3,5) [1, 6, 46, 396, 3606, 33876, 324556, 3151896, 30915046, 305543556]
(r,k) = (4,5) [1, 6, 46, 396, 3606, 33876, 324556, 3151896, 30915046, 305543556]
(r,k) = (5,5) [1, 6, 46, 396, 3606, 33876, 324556, 3151896, 30915046, 305543556]
(r,k) = (-5,6) [1, 7, 61, 595, 6145, 65527, 712909, 7863667, 87615745, 983726695]
(r,k) = (-4,6) [1, 7, 61, 595, 6145, 65527, 712909, 7863667, 87615745, 983726695]
(r,k) = (-3,6) [1, 7, 61, 595, 6145, 65527, 712909, 7863667, 87615745, 983726695]
(r,k) = (-2,6) [1, 7, 61, 595, 6145, 65527, 712909, 7863667, 87615745, 983726695]
(r,k) = (-1,6) [1, 7, 61, 595, 6145, 65527, 712909, 7863667, 87615745, 983726695]
(r,k) = (1,6) [1, 7, 61, 595, 6145, 65527, 712909, 7863667, 87615745, 983726695]
(r,k) = (2,6) [1, 7, 61, 595, 6145, 65527, 712909, 7863667, 87615745, 983726695]
(r,k) = (3,6) [1, 7, 61, 595, 6145, 65527, 712909, 7863667, 87615745, 983726695]
(r,k) = (4,6) [1, 7, 61, 595, 6145, 65527, 712909, 7863667, 87615745, 983726695]
(r,k) = (5,6) [1, 7, 61, 595, 6145, 65527, 712909, 7863667, 87615745, 983726695]
(r,k) = (-5,7) [1, 8, 78, 848, 9766, 116208, 1411404, 17383584, 216294534, 2712176048]
(r,k) = (-4,7) [1, 8, 78, 848, 9766, 116208, 1411404, 17383584, 216294534, 2712176048]
(r,k) = (-3,7) [1, 8, 78, 848, 9766, 116208, 1411404, 17383584, 216294534, 2712176048]
(r,k) = (-2,7) [1, 8, 78, 848, 9766, 116208, 1411404, 17383584, 216294534, 2712176048]
(r,k) = (-1,7) [1, 8, 78, 848, 9766, 116208, 1411404, 17383584, 216294534, 2712176048]
(r,k) = (1,7) [1, 8, 78, 848, 9766, 116208, 1411404, 17383584, 216294534, 2712176048]
(r,k) = (2,7) [1, 8, 78, 848, 9766, 116208, 1411404, 17383584, 216294534, 2712176048]
(r,k) = (3,7) [1, 8, 78, 848, 9766, 116208, 1411404, 17383584, 216294534, 2712176048]
(r,k) = (4,7) [1, 8, 78, 848, 9766, 116208, 1411404, 17383584, 216294534, 2712176048]
(r,k) = (5,7) [1, 8, 78, 848, 9766, 116208, 1411404, 17383584, 216294534, 2712176048]
(r,k) = (-5,8) [1, 9, 97, 1161, 14721, 192969, 2582881, 35066313, 481003009, 6649718409]
(r,k) = (-4,8) [1, 9, 97, 1161, 14721, 192969, 2582881, 35066313, 481003009, 6649718409]
(r,k) = (-3,8) [1, 9, 97, 1161, 14721, 192969, 2582881, 35066313, 481003009, 6649718409]
(r,k) = (-2,8) [1, 9, 97, 1161, 14721, 192969, 2582881, 35066313, 481003009, 6649718409]
(r,k) = (-1,8) [1, 9, 97, 1161, 14721, 192969, 2582881, 35066313, 481003009, 6649718409]
(r,k) = (1,8) [1, 9, 97, 1161, 14721, 192969, 2582881, 35066313, 481003009, 6649718409]
(r,k) = (2,8) [1, 9, 97, 1161, 14721, 192969, 2582881, 35066313, 481003009, 6649718409]
(r,k) = (3,8) [1, 9, 97, 1161, 14721, 192969, 2582881, 35066313, 481003009, 6649718409]
(r,k) = (4,8) [1, 9, 97, 1161, 14721, 192969, 2582881, 35066313, 481003009, 6649718409]
(r,k) = (5,8) [1, 9, 97, 1161, 14721, 192969, 2582881, 35066313, 481003009, 6649718409]
(r,k) = (-5,9) [1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300]
(r,k) = (-4,9) [1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300]
(r,k) = (-3,9) [1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300]
(r,k) = (-2,9) [1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300]
(r,k) = (-1,9) [1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300]
(r,k) = (1,9) [1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300]
(r,k) = (2,9) [1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300]
(r,k) = (3,9) [1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300]
(r,k) = (4,9) [1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300]
(r,k) = (5,9) [1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300]
(r,k) = (-5,10) [1, 11, 141, 1991, 29761, 460251, 7272861, 116619591, 1889815041, 30869546411]
(r,k) = (-4,10) [1, 11, 141, 1991, 29761, 460251, 7272861, 116619591, 1889815041, 30869546411]
(r,k) = (-3,10) [1, 11, 141, 1991, 29761, 460251, 7272861, 116619591, 1889815041, 30869546411]
(r,k) = (-2,10) [1, 11, 141, 1991, 29761, 460251, 7272861, 116619591, 1889815041, 30869546411]
(r,k) = (-1,10) [1, 11, 141, 1991, 29761, 460251, 7272861, 116619591, 1889815041, 30869546411]
(r,k) = (1,10) [1, 11, 141, 1991, 29761, 460251, 7272861, 116619591, 1889815041, 30869546411]
(r,k) = (2,10) [1, 11, 141, 1991, 29761, 460251, 7272861, 116619591, 1889815041, 30869546411]
(r,k) = (3,10) [1, 11, 141, 1991, 29761, 460251, 7272861, 116619591, 1889815041, 30869546411]
(r,k) = (4,10) [1, 11, 141, 1991, 29761, 460251, 7272861, 116619591, 1889815041, 30869546411]
(r,k) = (5,10) [1, 11, 141, 1991, 29761, 460251, 7272861, 116619591, 1889815041, 30869546411]
(00:00) gp >

200719

Python


棒倒し法で迷路生成

「棒を格子状に配置し、倒す」というシンプルなアルゴリズムなので、

簡単に実装できる。

import random

class Maze():
    """ 迷路を作るクラス"""
    # 壁は1、道は0
    PATH = 0
    WALL = 1

    def __init__(self, col, row):
        self.maze = []
        self.col = col
        self.row = row
        self.start_X = 1
        self.start_Y = 1
        self.goal_X = col - 2
        self.goal_Y = row - 2

        # 迷路は、幅および高さ5以上の奇数で生成する。
        if(self.row < 5 or self.col < 5):
            print('at least 5...')
            exit()
        if (self.col % 2) == 0:
            self.col += 1
            self.goal_X += 1
        if (self.row % 2) == 0:
            self.row += 1
            self.goal_Y += 1

    def set_maze(self):
        """ 迷路を作る。"""
        for x in range(0, self.col):
            row = []
            for y in range(0, self.row):
                if (x == 0 or y == 0 or x == self.col - 1 or y == self.row - 1):
                    cell = self.WALL
                else:
                    cell = self.PATH
                row.append(cell)
            self.maze.append(row)

        self.maze[self.start_X][self.start_Y] = 'S'
        self.maze[self.goal_X][self.goal_Y] = 'G'

        # 棒を倒す向き
        points = [[0, -1], [0, 1], [1, 0], [-1, 0]]

        # 一つおきに棒を配置
        for x in range(2, self.col - 1, 2):
            for y in range(2, self.row - 1, 2):
                self.maze[x][y] = self.WALL

                while True:
                    if x == 2:
                        r = points[random.randrange(0, 4)]
                    else:
                        r = points[random.randrange(0, 3)]

                    # 壁にする方向が壁でない場合は壁にする。
                    if self.maze[x + r[0]][y + r[1]] != self.WALL:
                        self.maze[x + r[0]][y + r[1]] = self.WALL
                        break

        return self.maze

    def show_maze(self):
        """ 迷路を出力する。"""
        for row in self.maze:
            for cell in row:
                if cell == self.PATH:
                    print('  ', end='')
                elif cell == self.WALL:
                    print('##', end='')
                elif cell == 'S':
                    print('S ', end='')
                elif cell == 'G':
                    print(' G', end='')
            print()


maze = Maze(10, 15)
maze.set_maze()
maze.show_maze()

200711

PARI


A336163 とA336179

A268545 のGheorghe Coserea さんのコード参考に

diagonal of the rational function 1 / (1 + y + z + x*y + y*z + k*z*x + (k+1)*x*y*z)

と

diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) - k*x*y*z)

が一致することを確認してみた。

(21:46) gp > N=10;
(21:46) gp > diag(n, expr, var=variables(expr)) = {
  my(a=vector(n));
  for(i=1, #var, expr=taylor(expr, var[#var-i+1], n));
  for(j=1, n, a[j]=expr;
    for(i=1, #var, a[j]=polcoeff(a[j], j-1)));
  return(a);
};
(21:46) gp > R_1(k) = 1/(1+y+z+x*y+y*z+k*z*x+(k+1)*x*y*z);
(21:46) gp > R_2(k) = 1/((1-x)*(1-y)*(1-z)-k*x*y*z);
(21:46) gp > for(k=-10, 10, v_1=diag(N, R_1(k)); v_2=diag(N, R_2(k)); if(v_1==v_2, print(v_1)))
[1, -9, 21, 1431, -33039, 248751, 5485341, -206302329, 2626974081, 17422114311]
[1, -8, 10, 1216, -23174, 111952, 4359664, -118557440, 1014582682, 17402599792]
[1, -7, 1, 1001, -15359, 30233, 3126529, -61392247, 259448833, 11970181433]
[1, -6, -6, 792, -9414, -11556, 2010576, -27431424, -20457702, 6451563924]
[1, -5, -11, 595, -5135, -26525, 1133245, -9654125, -75662975, 2718080875]
[1, -4, -14, 416, -2294, -25624, 533296, -1992064, -52440134, 824947256]
[1, -3, -15, 261, -639, -17523, 188049, 284085, -21320703, 137339037]
[1, -2, -14, 136, 106, -8492, 35344, 395008, -4547462, -4838372]
[1, -1, -11, 47, 241, -2281, -3779, 104831, -110207, -4415281]
[1, 0, -6, 0, 90, 0, -1680, 0, 34650, 0]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[1, 2, 10, 56, 346, 2252, 15184, 104960, 739162, 5280932]
[1, 3, 21, 171, 1521, 14283, 138909, 1385163, 14072193, 145039923]
[1, 4, 34, 352, 3946, 46744, 573616, 7217536, 92527738, 1203467464]
[1, 5, 49, 605, 8065, 113525, 1656145, 24774125, 377601025, 5839329125]
[1, 6, 66, 936, 14346, 231876, 3885456, 66767616, 1169068986, 20769386796]
[1, 7, 85, 1351, 23281, 422527, 7951069, 153458935, 3018043777, 60225528727]
[1, 8, 106, 1856, 35386, 709808, 14762224, 314931968, 6846486298, 151025959568]
[1, 9, 129, 2457, 51201, 1121769, 25477761, 593640441, 14095941633, 339631646409]
[1, 10, 154, 3160, 71290, 1690300, 41536720, 1047232000, 26908298650, 701590424500]
[1, 11, 181, 3971, 96241, 2451251, 64689661, 1751654531, 48341416321, 1353802736891]
(21:46) gp >

200627

JavaScript


Sum_{n = 1..24} n^2 = 70^2(3)

Sum_{n = 1..24} n^2 = 70^2

と

Sum_{n = 15..34} n^3 = 70^3

に関するHTMLを書いてみた。

<!DOCTYPE html>
<html>
  <head>
    <meta charset='utf-8'>
    <title>Sum_{n = 1..24} n^2 = 70^2</title>
  </head>
  <body>
    <script>
      function showSum(from, to, power){
        var sum = from ** power;
        var formula = from + '^' + power;
        for(var i=from + 1; i<=to + 1; i++){
          document.write('<p>' + formula + ' = ' + sum + '</p>');
          sum += i ** power;
          formula += '+' + i + '^' + power;
        }
      }
      showSum(1, 24, 2)
      showSum(15, 34, 3)
    </script>
  </body>
</html>

200620

Ruby


1 / n の性質について(4)

n が99...9 の約数のとき、

1 / n の小数表示が面白い。

require 'bigdecimal'

def A(n)
  puts "1/#{n}"
  if n == 1
    print "= 0." + "9" * 50
  else
    print "= " + BigDecimal(1r / n, 50).to_s("F")[0..51]
  end
  puts "..."
end

def B(n)
  m = 10 ** n - 1
  (1..Math.sqrt(m)).each{|i|
    if m % i == 0
      A(i)
      A(m / i)
    end
  }
end

(1..7).each{|i| B(i)}

出力結果

1/1

= 0.99999999999999999999999999999999999999999999999999...

1/9

= 0.11111111111111111111111111111111111111111111111111...

1/3

= 0.33333333333333333333333333333333333333333333333333...

1/3

= 0.33333333333333333333333333333333333333333333333333...

1/1

= 0.99999999999999999999999999999999999999999999999999...

1/99

= 0.01010101010101010101010101010101010101010101010101...

1/3

= 0.33333333333333333333333333333333333333333333333333...

1/33

= 0.03030303030303030303030303030303030303030303030303...

1/9

= 0.11111111111111111111111111111111111111111111111111...

1/11

= 0.09090909090909090909090909090909090909090909090909...

1/1

= 0.99999999999999999999999999999999999999999999999999...

1/999

= 0.00100100100100100100100100100100100100100100100100...

1/3

= 0.33333333333333333333333333333333333333333333333333...

1/333

= 0.00300300300300300300300300300300300300300300300300...

1/9

= 0.11111111111111111111111111111111111111111111111111...

1/111

= 0.00900900900900900900900900900900900900900900900900...

1/27

= 0.03703703703703703703703703703703703703703703703703...

1/37

= 0.02702702702702702702702702702702702702702702702702...

1/1

= 0.99999999999999999999999999999999999999999999999999...

1/9999

= 0.00010001000100010001000100010001000100010001000100...

1/3

= 0.33333333333333333333333333333333333333333333333333...

1/3333

= 0.00030003000300030003000300030003000300030003000300...

1/9

= 0.11111111111111111111111111111111111111111111111111...

1/1111

= 0.00090009000900090009000900090009000900090009000900...

1/11

= 0.09090909090909090909090909090909090909090909090909...

1/909

= 0.00110011001100110011001100110011001100110011001100...

1/33

= 0.03030303030303030303030303030303030303030303030303...

1/303

= 0.00330033003300330033003300330033003300330033003300...

1/99

= 0.01010101010101010101010101010101010101010101010101...

1/101

= 0.00990099009900990099009900990099009900990099009900...

1/1

= 0.99999999999999999999999999999999999999999999999999...

1/99999

= 0.00001000010000100001000010000100001000010000100001...

1/3

= 0.33333333333333333333333333333333333333333333333333...

1/33333

= 0.00003000030000300003000030000300003000030000300003...

1/9

= 0.11111111111111111111111111111111111111111111111111...

1/11111

= 0.00009000090000900009000090000900009000090000900009...

1/41

= 0.02439024390243902439024390243902439024390243902439...

1/2439

= 0.00041000410004100041000410004100041000410004100041...

1/123

= 0.00813008130081300813008130081300813008130081300813...

1/813

= 0.00123001230012300123001230012300123001230012300123...

1/271

= 0.00369003690036900369003690036900369003690036900369...

1/369

= 0.00271002710027100271002710027100271002710027100271...

1/1

= 0.99999999999999999999999999999999999999999999999999...

1/999999

= 0.00000100000100000100000100000100000100000100000100...

1/3

= 0.33333333333333333333333333333333333333333333333333...

1/333333

= 0.00000300000300000300000300000300000300000300000300...

1/7

= 0.14285714285714285714285714285714285714285714285714...

1/142857

= 0.00000700000700000700000700000700000700000700000700...

1/9

= 0.11111111111111111111111111111111111111111111111111...

1/111111

= 0.00000900000900000900000900000900000900000900000900...

1/11

= 0.09090909090909090909090909090909090909090909090909...

1/90909

= 0.00001100001100001100001100001100001100001100001100...

1/13

= 0.07692307692307692307692307692307692307692307692307...

1/76923

= 0.00001300001300001300001300001300001300001300001300...

1/21

= 0.04761904761904761904761904761904761904761904761904...

1/47619

= 0.00002100002100002100002100002100002100002100002100...

1/27

= 0.03703703703703703703703703703703703703703703703703...

1/37037

= 0.00002700002700002700002700002700002700002700002700...

1/33

= 0.03030303030303030303030303030303030303030303030303...

1/30303

= 0.00003300003300003300003300003300003300003300003300...

1/37

= 0.02702702702702702702702702702702702702702702702702...

1/27027

= 0.00003700003700003700003700003700003700003700003700...

1/39

= 0.02564102564102564102564102564102564102564102564102...

1/25641

= 0.00003900003900003900003900003900003900003900003900...

1/63

= 0.01587301587301587301587301587301587301587301587301...

1/15873

= 0.00006300006300006300006300006300006300006300006300...

1/77

= 0.01298701298701298701298701298701298701298701298701...

1/12987

= 0.00007700007700007700007700007700007700007700007700...

1/91

= 0.01098901098901098901098901098901098901098901098901...

1/10989

= 0.00009100009100009100009100009100009100009100009100...

1/99

= 0.01010101010101010101010101010101010101010101010101...

1/10101

= 0.00009900009900009900009900009900009900009900009900...

1/111

= 0.00900900900900900900900900900900900900900900900900...

1/9009

= 0.00011100011100011100011100011100011100011100011100...

1/117

= 0.00854700854700854700854700854700854700854700854700...

1/8547

= 0.00011700011700011700011700011700011700011700011700...

1/143

= 0.00699300699300699300699300699300699300699300699300...

1/6993

= 0.00014300014300014300014300014300014300014300014300...

1/189

= 0.00529100529100529100529100529100529100529100529100...

1/5291

= 0.00018900018900018900018900018900018900018900018900...

1/231

= 0.00432900432900432900432900432900432900432900432900...

1/4329

= 0.00023100023100023100023100023100023100023100023100...

1/259

= 0.00386100386100386100386100386100386100386100386100...

1/3861

= 0.00025900025900025900025900025900025900025900025900...

1/273

= 0.00366300366300366300366300366300366300366300366300...

1/3663

= 0.00027300027300027300027300027300027300027300027300...

1/297

= 0.00336700336700336700336700336700336700336700336700...

1/3367

= 0.00029700029700029700029700029700029700029700029700...

1/333

= 0.00300300300300300300300300300300300300300300300300...

1/3003

= 0.00033300033300033300033300033300033300033300033300...

1/351

= 0.00284900284900284900284900284900284900284900284900...

1/2849

= 0.00035100035100035100035100035100035100035100035100...

1/407

= 0.00245700245700245700245700245700245700245700245700...

1/2457

= 0.00040700040700040700040700040700040700040700040700...

1/429

= 0.00233100233100233100233100233100233100233100233100...

1/2331

= 0.00042900042900042900042900042900042900042900042900...

1/481

= 0.00207900207900207900207900207900207900207900207900...

1/2079

= 0.00048100048100048100048100048100048100048100048100...

1/693

= 0.00144300144300144300144300144300144300144300144300...

1/1443

= 0.00069300069300069300069300069300069300069300069300...

1/777

= 0.00128700128700128700128700128700128700128700128700...

1/1287

= 0.00077700077700077700077700077700077700077700077700...

1/819

= 0.00122100122100122100122100122100122100122100122100...

1/1221

= 0.00081900081900081900081900081900081900081900081900...

1/999

= 0.00100100100100100100100100100100100100100100100100...

1/1001

= 0.00099900099900099900099900099900099900099900099900...

1/1

= 0.99999999999999999999999999999999999999999999999999...

1/9999999

= 0.00000010000001000000100000010000001000000100000010...

1/3

= 0.33333333333333333333333333333333333333333333333333...

1/3333333

= 0.00000030000003000000300000030000003000000300000030...

1/9

= 0.11111111111111111111111111111111111111111111111111...

1/1111111

= 0.00000090000009000000900000090000009000000900000090...

1/239

= 0.00418410041841004184100418410041841004184100418410...

1/41841

= 0.00002390000239000023900002390000239000023900002390...

1/717

= 0.00139470013947001394700139470013947001394700139470...

1/13947

= 0.00007170000717000071700007170000717000071700007170...

1/2151

= 0.00046490004649000464900046490004649000464900046490...

1/4649

= 0.00021510002151000215100021510002151000215100021510...