151020

Ruby

Number of times k is used in writing out all the numbers 1 through n(9)

Number of times k is used in writing out all the numbers 1 through n(4)
のコードを工夫してみた。

# 高速化（ただし、0 < k ≦ m - 1）
def g(m, k, str, s)
  if s == 1
    return 0 if str.to_i < k
    return 1
  end
  a = str[0].to_i
  str1 = str[1..-1]
  if a < k
    b = 0
  elsif a == k
    b = 1 + str1.to_i(m)
  else
    b = m ** (s - 1)
  end
  return a * (s - 1) * m ** (s - 2) + b + g(m, k, str1, s - 1)
end

# 0 < k ≦ m - 1におけるg(m, k, str, s)の和
def g_sum(m, str, s)
  return str.to_i if s == 1
  a = str[0].to_i
  str1 = str[1..-1]
  b = 0
  b += 1 + str1.to_i(m) + (a - 1) * m ** (s - 1) if a > 0
  return (m - 1) * a * (s - 1) * m ** (s - 2) + b + g_sum(m, str1, s - 1)
end

# 1からnまでをm進法で表したときの桁数の合計
def g_total(m, n)
  s = n.to_s(m).size
  (n - m ** (s - 1) + 1) * s + ((s - 1) * m ** s - s * m ** (s - 1) + 1) / (m - 1)
end

# 0 ≦ k ≦ m - 1
def f(m, k, n)
  str = n.to_s(m)
  return g(m, k, str, str.size) if k > 0
  # k = 0のときは、桁数の合計から0以外の数字が現れた回数を除く
  return g_total(m, n) - g_sum(m, str, str.size)
end

def f_ary(m, k, ary)
  ary.map{|i| f(m, k, i)}
end

ary = (0..25).map{|i| 10 ** i}
(2..10).each{|m|
  p [m, 'total']
  p ary.map{|n| g_total(m, n)}
  (0..m - 1).each{|k|
    p [m, k]
    p f_ary(m, k, ary)
  }
}

出力結果
[2, "total"]
[1, 29, 580, 8987, 123631, 1568946, 18951445, 223222809, 2565782300, 28926258207, 322820130851, 3562561046566, 38900488372265, 422407813955629, 4559262511644720, 48874100093157427, 521985601490518071, 5555884811924144186, 58847078495393153085, 621553255926290448449, 6552426047410323587140, 68819408379282588696647, 721110534068521419145291, 7548884272548171353161806, 78791074180385370825293905, 820657186886165933204701269]
[2, 0]
[0, 12, 261, 4049, 59018, 753916, 9066446, 108788177, 1251335184, 14079330066, 158527003672, 1752835390487, 19090546253852, 208098347208735, 2252849861951519, 24089352692482079, 257698738277646371, 2751668512654557222, 29085855711963906085, 307321301701485658158, 3250747955772179611690, 34097225207936917176362, 356740182741316423122994, 3749253699468361851732019, 39095688742747483198718003, 406175259971825928562016322]
[2, 1]
[1, 17, 319, 4938, 64613, 815030, 9884999, 114434632, 1314447116, 14846928141, 164293127179, 1809725656079, 19809942118413, 214309466746894, 2306412649693201, 24784747400675348, 264286863212871700, 2804216299269586964, 29761222783429247000, 314231954224804790291, 3301678091638143975450, 34722183171345671520285, 364370351327204996022297, 3799630573079809501429787, 39695385437637887626575902, 414481926914340004642684947]
[3, "total"]
[1, 20, 384, 5914, 80168, 1011438, 12202852, 142825562, 1635429936, 18418869286, 204769823420, 2258785231784, 24729067085862, 268561603772548, 2897054433952706, 31073489905574112, 331661409150166750, 3524952682351500476, 37324574141163503994, 393921167270471535640, 4145290505434243820438, 43522843646724583150724, 456705592820521248356154, 4780350335384691235205008, 49923153018462221116844678, 520308377166159990051601692]
[3, 0]
[0, 5, 112, 1830, 24856, 312299, 3780509, 44933595, 521626139, 5880056822, 65288716800, 729403871260, 8069935693754, 87955158771471, 950747326623765, 10126147772237068, 108084168326741432, 1141785638343655929, 12192227578518173960, 129047966872781701145, 1353915031388354912698, 14214508943285390237646, 150454979304746373557149, 1577741974974853061914468, 16498568363160802749548694, 171387305141808329923338624]
[3, 1]
[1, 9, 150, 2194, 30374, 379432, 4508710, 51180200, 572158742, 6417405502, 70402455152, 769364418852, 8452692931320, 91673681872620, 993534122918024, 10779360089419044, 114661242594171254, 1216758202087170334, 12767569182895088152, 133862436956018539886, 1405771200630725286006, 14663860400505783919608, 153796494936787179989436, 1614986483014523055902106, 16887592497038459726705624, 177308495497250974401699424]
[3, 2]
[0, 6, 122, 1890, 24938, 319707, 3913633, 46711767, 541645055, 6121406962, 69078651468, 760016941672, 8206438460788, 88932763128457, 952772984410917, 10167982043918000, 108915998229254064, 1166408841920674213, 12364777379750241882, 131010763441671294609, 1385604273415163621734, 14644474302933408993470, 152454118578987694809569, 1587621877395315117388434, 16536992158262958640590360, 171612576527100685726563644]
[4, "total"]
[1, 17, 319, 4664, 64546, 812628, 9650485, 114407607, 1310521529, 14642086074, 164273376956, 1808374031038, 19633496124095, 214135937985217, 2306175007763139, 24624700031052484, 263995200496839366, 2803923207949429448, 29615692831797717705, 313851085308763482827, 3301617364940215724749, 34606469459760862898894, 363703511356173806381776, 3799256181698780902107858, 39597024726795123608431315, 413552395628721977734900437]
[4, 0]
[0, 2, 65, 930, 14294, 187918, 2187936, 26908111, 311721392, 3432994224, 39525887923, 438139377842, 4694047261364, 51963626752693, 563444246338230, 5950299292388021, 64318496246508216, 689832613531986615, 7196883791014767287, 76667563571579759292, 815418493853503761081, 8458340839333464550072, 89018412185597045811900, 941665524969631256324797, 9715438621335639516949183, 101390308675072882998160066]
[4, 1]
[1, 7, 119, 1270, 19482, 242178, 2520434, 31973411, 366171397, 3790564097, 44245876996, 488164976133, 5056725669893, 56692742610948, 610964919316485, 6310200205451268, 69321688173772803, 732713250404499465, 7571981434117226501, 82037556623093661699, 854087328285483073545, 8814800381086001004552, 94722289936057215483909, 973215603996092086091784, 10042775926285679481323524, 107193424611029871262433293]
[4, 2]
[0, 6, 70, 1260, 16409, 192212, 2519665, 28590949, 316981761, 3780691462, 40955973891, 442578674180, 5012229937156, 53342413107206, 568083953840132, 6253853871202310, 65744857114017797, 691251639160799237, 7504413864019427335, 78097532626468995080, 816632322249796878341, 8759273688423124172807, 90277557077891309633546, 942335136382397704044551, 10025010846995401075064838, 102648822381928313568362502]
[4, 3]
[0, 2, 65, 1204, 14361, 190320, 2422450, 26935136, 315646979, 3637836291, 39545638146, 439491002883, 4870493255682, 52137155514370, 563681888268292, 6110346662010885, 64610158962540550, 690125704852144131, 7342413742646296582, 77048432487621066756, 815479220551432011782, 8574054550918273171463, 89685252156628235452421, 942039916350659855646726, 9813799332178403535093770, 102319839960690909905944576]
[5, "total"]
[1, 16, 272, 4224, 56100, 702352, 8511728, 97792980, 1138964856, 12694824232, 142370605484, 1561853027360, 17046325683612, 185231628417988, 1980790710449240, 21403953552246116, 227019767761230492, 2425494194030761744, 25627470970153808620, 270686774253845214872, 2853433871269226074248, 29835846781730651855500, 314179233908653259277376, 3270896169543266296386752, 34272404238581657409668004, 356362021192908287048339880]
[5, 0]
[0, 2, 36, 697, 9723, 133599, 1539850, 17964853, 211464853, 2363964854, 27106933605, 297575683605, 3329870605481, 35417419433608, 383416442871110, 4130332946777361, 43783256530761736, 470461692810058611, 4967175102233886738, 53233740806579589863, 554061484336853027366, 5848109316825866699243, 61451171278953552246119, 638078004717826843261742, 6697390648722648620605493, 69689764007925987243652371]
[5, 1]
[1, 7, 65, 1226, 13001, 165627, 2028125, 20171876, 268203127, 2645703125, 32294921876, 331865234377, 3689306640628, 40104003906250, 403492431640629, 4679504394531250, 46297943115234377, 525622406005859377, 5326352691650390626, 57214763641357421878, 600465202331542968751, 6073609542846679687504, 66583832740783691406250, 664710590839385986328129, 7260833132266998291015627, 73223027288913726806640626]
[5, 2]
[0, 3, 65, 850, 13000, 134376, 1840627, 19937500, 221328126, 2637890627, 28398437500, 328398437500, 3360693359376, 38648925781253, 397962646484375, 4234832763671878, 46295654296875001, 482975616455078127, 5279622650146484376, 53592174530029296877, 587362003326416015628, 5973327350616455078127, 62607814311981201171878, 661748609542846679687501, 6841735947132110595703127, 72742526531219482421875006]
[5, 3]
[0, 2, 65, 726, 10501, 134375, 1559375, 19859376, 218984375, 2633984375, 27412109376, 306279296877, 3336279296876, 35572753906252, 397962646484375, 4179687500000002, 46295501708984375, 475630035400390627, 5086095428466796878, 53411624908447265628, 557347488403320312503, 5973314476013183593750, 61840394973754882812502, 661399910449981689453127, 6771569001674652099609381, 70993635952472686767578126]
[5, 4]
[0, 2, 41, 725, 9875, 134375, 1543751, 19859375, 218984375, 2413281251, 27158203127, 297734375001, 3330175781251, 35488525390625, 397956542968751, 4179595947265625, 44347412109375003, 470804443359375002, 4968225097656250002, 53234470367431640626, 554197692871093750000, 5967486095428466796876, 61696020603179931640627, 644959053993225097656253, 6700875508785247802734376, 69713067412376403808593751]
[6, "total"]
[1, 15, 260, 3745, 50675, 644020, 7664085, 87984470, 1027440600, 11564643545, 127387861210, 1405963003100, 15435778018525, 166614668111070, 1779688008666335, 19268768311987425, 205612609871924450, 2173675659231546595, 23052323732335676645, 244313942394014059750, 2565883654364084358375, 26795301926184506150120, 282630869342642221403370, 2955785216055853328420075, 30734711296335119970520300, 320449606668064318938729710]
[6, 0]
[0, 1, 28, 472, 7792, 96602, 1135660, 13028618, 158363049, 1781669668, 19745872648, 227909003889, 2459990037720, 26379763779105, 280500235869852, 3085218547843454, 32833258280691482, 347683017078293947, 3782653531545490550, 39534227235118387853, 413735974024774674039, 4307326759431044312455, 45870435722012457184613, 478528913697009862151535, 4977878570157856366073921, 52862340470486562513187627]
[6, 1]
[1, 7, 71, 731, 11198, 145691, 1471583, 15044020, 209959005, 2215950297, 22357576982, 260732117894, 2964571630298, 29734482337166, 300804840123645, 3596403661788849, 37220596165405190, 373942198480119148, 4097855055721481385, 44696639571507276727, 447357911583038454820, 4507221089612289882735, 50957678725204529674292, 522423493113073904971028, 5237942923211234920118775, 55861845725776606242564250]
[6, 2]
[0, 2, 64, 731, 8015, 105724, 1468480, 15036280, 170401845, 2126695994, 22349164466, 230521001428, 2553734780882, 29686729820416, 300655135938110, 3205679939605408, 36193990212721421, 373397802196303303, 3808925198520288348, 40481495552137836674, 447357910133301379941, 4506949179057162814305, 47098247702572249166961, 511150472996604417121406, 5237921416451673325535601, 53086650812786108689970093]
[6, 3]
[0, 2, 35, 724, 8015, 99035, 1317036, 15036215, 169406949, 1844445489, 22150503153, 230377772246, 2492643173066, 28053573741924, 300332901443709, 3189692886768142, 33445910414365065, 370957646611641398, 3797461074214285846, 39955008728061630146, 429356627553235346921, 4506931335876576179484, 46855356118085672416370, 484840463353143769132156, 5206242491098553003089692, 52914044231686680108760939]
[6, 4]
[0, 2, 34, 615, 7863, 99034, 1135666, 15005175, 160043362, 1814212292, 21038871313, 228282586508, 2486112825675, 26380129900105, 300261067622732, 3103725613575301, 33085595600958522, 359974948542551848, 3782714438353511379, 39826535380747513568, 414043307158573756000, 4506927171526374932207, 45978714519912057008219, 480312952883968590039808, 5095703161752445949084183, 52862369054840906489032458]
[6, 5]
[0, 1, 28, 472, 7792, 97934, 1135660, 14834162, 159266390, 1781669805, 19745872648, 228140521135, 2478725570884, 26379988532354, 297133827668287, 3088047662406271, 32833259197782770, 347720046322636951, 3782714433980619137, 39820035926441414782, 414031923911160746654, 4459946390681058028934, 45870436554855255952915, 478528920012052785004142, 4979022733663356406618128, 52862356372487454895214343]
[7, "total"]
[1, 14, 246, 3604, 47204, 580398, 7039208, 83274408, 952920802, 10670445554, 117693118812, 1286962821206, 14208739748358, 154461178238416, 1661228247668816, 17728597733681610, 188100184135771162, 2006909022652786020, 21348363158569502014, 225438542109986513966, 2368069794769905597624, 24776488563389339183224, 259047939606077619976770, 2723335577242543339837228, 28463349040697803378860428, 296243443284884623652022822]
[7, 0]
[0, 1, 24, 380, 5647, 68892, 965000, 10787085, 125460792, 1401841015, 15507485037, 173892889695, 1948876827701, 21107004750063, 223612551309580, 2415999343094932, 25641007828765980, 281441739005408223, 2929553616259257873, 31040583783058389625, 325626422693525097264, 3419989862276212884294, 36231992403695785354266, 379802209504203819698306, 3985666563610435351361981, 41209487992549182629422359]
[7, 1]
[1, 6, 78, 780, 8110, 88500, 1219870, 15865600, 172539968, 1731378357, 17700998204, 189189870494, 2383125982820, 26133698711472, 261771343230968, 2684380968983890, 27311303300643582, 312355937793457211, 3463944584394735456, 35601953767056758691, 357533815140350476230, 3630857630320495382668, 37779610844512699452723, 425041907065853528823289, 4466458203133567081386951, 44618338243477807545984051]
[7, 2]
[0, 2, 32, 752, 8051, 88500, 984600, 11630400, 151004956, 1731277200, 17693302672, 184945218049, 2049485093130, 21884202483741, 261771115832026, 2654159875976645, 27310892224213832, 283308244555221459, 3022490448478019193, 33862717664619078382, 357524905452287316162, 3616096044778082900587, 37155361718610826104304, 390754201533875335193348, 4067256720922519257062602, 44615002147323622393703377]
[7, 3]
[0, 2, 28, 430, 8051, 88492, 974738, 11630383, 127478384, 1591931287, 17660643600, 184945215113, 1980278392686, 21377898645382, 228752947068295, 2651059378227450, 27310527974405944, 283106134049192629, 3019387088183460063, 31625362502476074870, 348049734186983053933, 3615136257368942300470, 36998957507338517202370, 387486491649978060196670, 3986792603742625997814037, 41735379124070902004799090]
[7, 4]
[0, 1, 28, 430, 6047, 88296, 965000, 11630347, 125480000, 1407624801, 17660299864, 184666031090, 1949167908510, 21377239972530, 228441196782410, 2487517888462580, 27310527973598975, 283106054946074662, 3011076142524411632, 31120620488588141543, 327962317935941536555, 3612131317529933102576, 36960930792918110573472, 380594594233042078452575, 3985834697584601982285263, 41395096385504451603020184]
[7, 5]
[0, 1, 28, 430, 5649, 87266, 965000, 10924285, 125480000, 1404549428, 15793795402, 184661800871, 1948922950960, 21300782832428, 228439874081111, 2417740139508408, 27310527942304340, 282144405942988784, 2972279068997955805, 31120608681853113819, 325686674703165126747, 3462287588768112743296, 36960543171334196001997, 379831996450583231248461, 3985670485000432091737956, 41340947588645677371268945]
[7, 6]
[0, 1, 28, 402, 5649, 70452, 965000, 10806308, 125476702, 1401843466, 15676594033, 184661795894, 1948882592551, 21280350842800, 228439219364426, 2417740139427705, 25905396891838509, 281446506360443052, 2929632209731661992, 31066695222334957036, 325685924657652990733, 3419989862347559869333, 36960543167667485287638, 379824176805007286224579, 3985669766703621617211638, 41329191803312980103824816]
[8, "total"]
[1, 13, 230, 3419, 45324, 562557, 6700414, 77603263, 880826048, 9846608321, 110182931907, 1221463455172, 13371707641285, 144973661130182, 1559789289041351, 16678314312330696, 177426514498645449, 1879412115989163466, 19835296927913307595, 209459003386451684813, 2215672027091613478350, 23325376216732907826639, 244603009733863262612944, 2556824077870906100903377, 26654592622967248807226834, 277236740983737990457814483]
[8, 0]
[0, 1, 20, 309, 4738, 59653, 721097, 8542809, 97787706, 1115927418, 13521667387, 145372195515, 1563478173115, 17331793272251, 183326684061116, 1978416700882364, 21105930005745085, 223633326877388220, 2356801045758897598, 25671791008780217793, 270629757607314419135, 2818137674520461798846, 29820969440903708241344, 309307602792579861278145, 3235625752192776180297152, 33690523215566065324748227]
[8, 1]
[1, 5, 66, 870, 9418, 95404, 1020178, 10910114, 116961408, 1269319104, 15420668802, 186335569154, 2092258707459, 22324231147523, 223533368487937, 2300102388453377, 23678250753204227, 244218428740583426, 2521303233634959362, 26987371554266808324, 307381704052157972481, 3312167967401039364099, 34662251320016253747201, 352132946710859256692738, 3580993929556662847275012, 36151550056345656205770756]
[8, 2]
[0, 2, 29, 381, 7124, 95373, 1020112, 10902625, 116961408, 1269319104, 13607942531, 155047253248, 1641958131714, 18648704704515, 217975554600962, 2300096106074114, 23675007796813825, 244187609128878083, 2521303226531643392, 26210740265861906435, 279711585926283001859, 2902447824095134089218, 30494440078365576134661, 338547063815532455133187, 3560961916499477187788801, 36125179996591085238353924]
[8, 3]
[0, 1, 29, 381, 5075, 63918, 971537, 10902241, 116953473, 1262686145, 13539799424, 151967442178, 1641152821249, 17340401500161, 188344290390017, 2170897047359490, 23541908635230209, 243975985889525762, 2521303173503582210, 26194734692110041089, 271945273057034108929, 2901583130260266549252, 29939173041882216792066, 314049571512541934256131, 3333146630949991994097667, 36125178266081207951294465]
[8, 4]
[0, 1, 26, 381, 4755, 62053, 754449, 10416353, 116928577, 1252198850, 13523801473, 146455220993, 1635778152450, 17333126389762, 188343491354627, 1993331687067651, 22106987706359810, 242568603890958340, 2519252240709910530, 26192725864872542208, 271945272702129668099, 2895818520444168568834, 29931803566205061038083, 314030922173988236951554, 3236253103937606641516546, 34035718740737068781010949]
[8, 5]
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[8, 6]
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[8, 7]
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[9, "total"]
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[9, 0]
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[9, 1]
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[9, 2]
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[9, 3]
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[9, 4]
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[9, 5]
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[9, 6]
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[9, 7]
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[9, 8]
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[10, "total"]
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[10, 0]
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[10, 1]
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[10, 2]
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[10, 3]
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[10, 4]
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[10, 5]
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[10, 6]
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[10, 7]
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[10, 8]
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[10, 9]
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