仕事の部屋

2021年7月31日土曜日

210731

Ruby

A278559

p(25*n+24) を出力してみた。

def s(k, n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0 && i % k == 0}
  s
end

def A(ary, n)
  a_ary = [1]
  a = [0] + (1..n).map{|i| ary.inject(0){|s, j| s + j[1] * s(j[0], i)}}
  (1..n).each{|i| a_ary << (1..i).inject(0){|s, j| s - a[j] * a_ary[-j]} / i}
  a_ary
end

n = 10
a = [5 ** 2 * 63, 5 ** 5 * 52, 5 ** 7 * 63, 5 ** 10 * 6, 5 ** 12]
ary = (0..4).map{|i| p [0] * i + A([[5, 6 * (i + 1)], [1, -6 * (i + 1) - 1]], n)}
(0..n).each{|i|
  print "#{(0..4).inject(0){|s, j| s + a[j] * ary[j][i]}} = "
  (0..4).each{|j|
    print " + " if j > 0
    print "#{a[j]} * #{ary[j][i]}"
  }
  puts
}

出力結果

[1, 7, 35, 140, 490, 1541, 4480, 12195, 31465, 77525, 183626]

[0, 1, 13, 104, 637, 3276, 14808, 60541, 228124, 803010, 2667054, 8422715]

[0, 0, 1, 19, 209, 1710, 11495, 66862, 347339, 1645875, 7221520, 29668595, 115116233]

[0, 0, 0, 1, 25, 350, 3575, 29575, 209381, 1312075, 7443825, 38854075, 188836375, 862496902]

[0, 0, 0, 0, 1, 31, 527, 6448, 63240, 526443, 3852742, 25380847, 153068700, 855816380, 4479330091]

1575 = 1575 * 1 + 162500 * 0 + 4921875 * 0 + 58593750 * 0 + 244140625 * 0

173525 = 1575 * 7 + 162500 * 1 + 4921875 * 0 + 58593750 * 0 + 244140625 * 0

7089500 = 1575 * 35 + 162500 * 13 + 4921875 * 1 + 58593750 * 0 + 244140625 * 0

169229875 = 1575 * 140 + 162500 * 104 + 4921875 * 19 + 58593750 * 1 + 244140625 * 0

2841940500 = 1575 * 490 + 162500 * 637 + 4921875 * 209 + 58593750 * 25 + 244140625 * 1

37027355200 = 1575 * 1541 + 162500 * 3276 + 4921875 * 1710 + 58593750 * 350 + 244140625 * 31

397125074750 = 1575 * 4480 + 162500 * 14808 + 4921875 * 11495 + 58593750 * 3575 + 244140625 * 527

3646072432125 = 1575 * 12195 + 162500 * 60541 + 4921875 * 66862 + 58593750 * 29575 + 244140625 * 6448

29454549941750 = 1575 * 31465 + 162500 * 228124 + 4921875 * 347339 + 58593750 * 209381 + 244140625 * 63240

213636919820625 = 1575 * 77525 + 162500 * 803010 + 4921875 * 1645875 + 58593750 * 1312075 + 244140625 * 526443

1412749565173450 = 1575 * 183626 + 162500 * 2667054 + 4921875 * 7221520 + 58593750 * 7443825 + 244140625 * 3852742

2021年6月26日土曜日

210626

OEISに載っている変わった数列

A027440 は何と「TV channels in Tokyo」である。
REFERENCES には

From a Tokyo kindergarten entrance examination.

と書いてある。

知識問題は、流石にプログラミングで答えが出せない。

2021年5月18日火曜日

210518

PARI

Sum_{n = 1..24} n^2 = 70^2(6)

n-gonal number であり、n-gonal pyramidal number である数について

比較的小さなもの（ただし0, 1 を除く）を出力してみた。

a(n) = for(k=2, n, if(ispolygonal(m=k*(k+1)*((n-2)*k-n+5)/6, n), print(n, ": ", [k, m])));
for(n=3, 100000, a(n))

出力結果

3: [3, 10]

10: [5, 175]

14: [6, 441]

30: [17, 23001]

43: [33, 245905]

50: [34, 314755]

88: [15, 48280]

276: [26, 801801]

322: [28, 1169686]

823: [113, 197427385]

2378: [103, 432684460]

2386: [420, 29437553530]

9525: [2169, 16195753597485]

16420: [6936, 913053565546276]

19605: [1191, 5519583702676]

31265: [259, 90525801730]

31368: [14858, 17147031694579605]

83135: [1310, 31148407558500]

2021年5月17日月曜日

210517

PARI

Sum_{n = 1..24} n^2 = 70^2(5)

n-gonal number であり、n-gonal pyramidal number である数を出力してみた。

a(n) = print1(n, ": "); for(k=0, 1e6, if(ispolygonal(m=k*(k+1)*((n-2)*k-n+5)/6, n), print1(m, ", "))); print;
for(n=3, 1000, a(n))

出力結果

3: 0, 1, 10, 120, 1540, 7140,

4: 0, 1, 4900,

5: 0, 1,

6: 0, 1, 946,

7: 0, 1,

8: 0, 1, 1045, 5985, 123395663059845, 774611255177760,

9: 0, 1,

10: 0, 1, 175, 368050005576,

11: 0, 1, 23725, 1519937678700, 7248070597636,

12: 0, 1,

13: 0, 1,

14: 0, 1, 441, 195661,

15: 0, 1,

16: 0, 1,

17: 0, 1, 975061, 1580765544996,

18: 0, 1,

19: 0, 1,

20: 0, 1, 3578401,

21: 0, 1,

22: 0, 1,

23: 0, 1, 10680265,

24: 0, 1,

25: 0, 1,

26: 0, 1, 27453385,

27: 0, 1,

28: 0, 1,

29: 0, 1, 63016921,

30: 0, 1, 23001,

31: 0, 1,

32: 0, 1, 132361021,

33: 0, 1,

34: 0, 1,

35: 0, 1, 258815701,

36: 0, 1,

37: 0, 1,

38: 0, 1, 477132085,

39: 0, 1,

40: 0, 1,

41: 0, 1, 55202400, 837244045,

42: 0, 1,

43: 0, 1, 245905,

44: 0, 1, 1408778281,

45: 0, 1,

46: 0, 1,

47: 0, 1, 2286380881,

48: 0, 1,

49: 0, 1,

50: 0, 1, 314755, 3595928401,

51: 0, 1,

52: 0, 1,

53: 0, 1, 5501691505,

54: 0, 1,

55: 0, 1,

56: 0, 1, 8214519205,

57: 0, 1,

58: 0, 1,

59: 0, 1, 12001111741,

60: 0, 1, 1785508245600,

61: 0, 1,

62: 0, 1, 17194450141,

63: 0, 1,

64: 0, 1,

65: 0, 1, 24205450501,

66: 0, 1,

67: 0, 1,

68: 0, 1, 33535911025,

69: 0, 1,

70: 0, 1,

71: 0, 1, 45792819865,

72: 0, 1,

73: 0, 1,

74: 0, 1, 61704091801,

75: 0, 1,

76: 0, 1,

77: 0, 1, 82135801801,

78: 0, 1,

79: 0, 1,

80: 0, 1, 108110983501,

81: 0, 1,

82: 0, 1,

83: 0, 1, 140830060645,

84: 0, 1,

85: 0, 1,

86: 0, 1, 181692979525,

87: 0, 1,

88: 0, 1, 48280,

89: 0, 1, 232323110461,

90: 0, 1,

91: 0, 1,

92: 0, 1, 294592986361,

93: 0, 1,

94: 0, 1,

95: 0, 1, 370651946401,

96: 0, 1,

97: 0, 1,

98: 0, 1, 462955752865,

99: 0, 1,

100: 0, 1,

101: 0, 1, 574298249185,

102: 0, 1,

103: 0, 1,

104: 0, 1, 707845127221,

105: 0, 1,

106: 0, 1,

107: 0, 1, 867169871821,

108: 0, 1,

109: 0, 1,

110: 0, 1, 1056291950701,

111: 0, 1,

112: 0, 1,

113: 0, 1, 1279717317685,

114: 0, 1,

115: 0, 1,

116: 0, 1, 1542481297345,

117: 0, 1,

118: 0, 1,

119: 0, 1, 1850193919081,

120: 0, 1,

121: 0, 1,

122: 0, 1, 2209087768681,

123: 0, 1,

124: 0, 1,

125: 0, 1, 2626068425401,

126: 0, 1,

127: 0, 1,

128: 0, 1, 3108767552605,

129: 0, 1,

130: 0, 1,

131: 0, 1, 3665598710005,

132: 0, 1,

133: 0, 1,

134: 0, 1, 4305815955541,

135: 0, 1,

136: 0, 1,

137: 0, 1, 5039575304941,

138: 0, 1,

139: 0, 1,

140: 0, 1, 5877999117001,

141: 0, 1,

142: 0, 1,

143: 0, 1, 6833243472625,

144: 0, 1,

145: 0, 1, 101337426,

146: 0, 1, 7918568615665,

147: 0, 1,

148: 0, 1,

149: 0, 1, 9148412523601,

150: 0, 1,

151: 0, 1,

152: 0, 1, 10538467676101,

153: 0, 1,

154: 0, 1,

155: 0, 1, 12105761089501,

156: 0, 1,

157: 0, 1,

158: 0, 1, 13868737685245,

159: 0, 1,

160: 0, 1,

161: 0, 1, 15847347060325,

162: 0, 1,

163: 0, 1,

164: 0, 1, 18063133727761,

165: 0, 1,

166: 0, 1,

167: 0, 1, 20539330895161,

168: 0, 1,

169: 0, 1,

170: 0, 1, 23300957849401,

171: 0, 1,

172: 0, 1,

173: 0, 1, 26374921015465,

174: 0, 1,

175: 0, 1,

176: 0, 1, 29790118757485,

177: 0, 1,

178: 0, 1,

179: 0, 1, 33577549990021,

180: 0, 1,

181: 0, 1,

182: 0, 1, 37770426667621,

183: 0, 1,

184: 0, 1,

185: 0, 1, 42404290220701,

186: 0, 1,

187: 0, 1,

188: 0, 1, 47517132005785,

189: 0, 1,

190: 0, 1,

191: 0, 1, 53149517838145,

192: 0, 1,

193: 0, 1,

194: 0, 1, 59344716674881,

195: 0, 1,

196: 0, 1,

197: 0, 1, 66148833516481,

198: 0, 1,

199: 0, 1,

200: 0, 1, 73610946594901,

201: 0, 1,

202: 0, 1,

203: 0, 1, 81783248916205,

204: 0, 1,

205: 0, 1,

206: 0, 1, 90721194225805,

207: 0, 1,

208: 0, 1,

209: 0, 1, 100483647464341,

210: 0, 1,

211: 0, 1,

212: 0, 1, 111133039782241,

213: 0, 1,

214: 0, 1,

215: 0, 1, 122735528181001,

216: 0, 1,

217: 0, 1,

218: 0, 1, 135361159849225,

219: 0, 1,

220: 0, 1,

221: 0, 1, 149084041261465,

222: 0, 1,

223: 0, 1,

224: 0, 1, 163982512107901,

225: 0, 1,

226: 0, 1,

227: 0, 1, 180139324122901,

228: 0, 1,

229: 0, 1,

230: 0, 1, 197641824880501,

231: 0, 1,

232: 0, 1,

233: 0, 1, 216582146624845,

234: 0, 1,

235: 0, 1,

236: 0, 1, 237057400203625,

237: 0, 1,

238: 0, 1,

239: 0, 1, 259169874172561,

240: 0, 1,

241: 0, 1,

242: 0, 1, 283027239138961,

243: 0, 1,

244: 0, 1,

245: 0, 1, 308742757412401,

246: 0, 1,

247: 0, 1,

248: 0, 1, 336435498030565,

249: 0, 1,

250: 0, 1,

251: 0, 1, 366230557228285,

252: 0, 1,

253: 0, 1,

254: 0, 1, 398259284417821,

255: 0, 1,

256: 0, 1,

257: 0, 1, 432659513748421,

258: 0, 1,

259: 0, 1,

260: 0, 1, 469575801313201,

261: 0, 1,

262: 0, 1,

263: 0, 1, 509159668071385,

264: 0, 1,

265: 0, 1,

266: 0, 1, 551569848553945,

267: 0, 1,

268: 0, 1,

269: 0, 1, 596972545420681,

270: 0, 1,

271: 0, 1,

272: 0, 1, 645541689936781,

273: 0, 1,

274: 0, 1,

275: 0, 1, 697459208436901,

276: 0, 1, 801801,

277: 0, 1,

278: 0, 1, 752915294844805,

279: 0, 1,

280: 0, 1,

281: 0, 1, 812108689316605,

282: 0, 1,

283: 0, 1,

284: 0, 1, 875246963075641,

285: 0, 1,

286: 0, 1,

287: 0, 1, 942546809507041,

288: 0, 1,

289: 0, 1,

290: 0, 1, 1014234341580001,

291: 0, 1,

292: 0, 1,

293: 0, 1, 1090545395665825,

294: 0, 1,

295: 0, 1,

296: 0, 1, 1171725841819765,

297: 0, 1,

298: 0, 1,

299: 0, 1, 1258031900594701,

300: 0, 1,

301: 0, 1,

302: 0, 1, 1349730466454701,

303: 0, 1,

304: 0, 1,

305: 0, 1, 1447099437856501,

306: 0, 1,

307: 0, 1,

308: 0, 1, 1550428054066945,

309: 0, 1,

310: 0, 1,

311: 0, 1, 1660017238784425,

312: 0, 1,

313: 0, 1,

314: 0, 1, 1776179950632361,

315: 0, 1,

316: 0, 1,

317: 0, 1, 1899241540592761,

318: 0, 1,

319: 0, 1,

320: 0, 1, 2029540116447901,

321: 0, 1,

322: 0, 1, 1169686,

323: 0, 1, 2167426914298165,

324: 0, 1,

325: 0, 1,

326: 0, 1, 2313266677224085,

327: 0, 1,

328: 0, 1,

329: 0, 1, 2467438041160621,

330: 0, 1,

331: 0, 1,

332: 0, 1, 2630333928051721,

333: 0, 1,

334: 0, 1,

335: 0, 1, 2802361946353201,

336: 0, 1,

337: 0, 1,

338: 0, 1, 2983944798951985,

339: 0, 1,

340: 0, 1,

341: 0, 1, 3175520698569745,

342: 0, 1,

343: 0, 1,

344: 0, 1, 3377543790718981,

345: 0, 1,

346: 0, 1,

347: 0, 1, 3590484584279581,

348: 0, 1,

349: 0, 1,

350: 0, 1, 3814830389763901,

351: 0, 1,

352: 0, 1,

353: 0, 1, 4051085765338405,

354: 0, 1,

355: 0, 1,

356: 0, 1, 4299772970669905,

357: 0, 1,

358: 0, 1,

359: 0, 1, 4561432428664441,

360: 0, 1,

361: 0, 1,

362: 0, 1, 4836623195166841,

363: 0, 1,

364: 0, 1,

365: 0, 1, 5125923436689001,

366: 0, 1,

367: 0, 1,

368: 0, 1, 5429930916234925,

369: 0, 1,

370: 0, 1,

371: 0, 1, 5749263487290565,

372: 0, 1,

373: 0, 1,

374: 0, 1, 15064335000, 6084559596046501,

375: 0, 1,

376: 0, 1,

377: 0, 1, 6436478791921501,

378: 0, 1,

379: 0, 1,

380: 0, 1, 6805702246455001,

381: 0, 1,

382: 0, 1,

383: 0, 1, 7192933280636545,

384: 0, 1,

385: 0, 1,

386: 0, 1, 7598897900740225,

387: 0, 1,

388: 0, 1,

389: 0, 1, 8024345342732161,

390: 0, 1,

391: 0, 1,

392: 0, 1, 8470048625319061,

393: 0, 1,

394: 0, 1,

395: 0, 1, 8936805111705901,

396: 0, 1,

397: 0, 1,

398: 0, 1, 9425437080130765,

399: 0, 1,

400: 0, 1,

401: 0, 1, 9936792303244885,

402: 0, 1,

403: 0, 1,

404: 0, 1, 10471744636405921,

405: 0, 1,

406: 0, 1,

407: 0, 1, 11031194614952521,

408: 0, 1,

409: 0, 1,

410: 0, 1, 11616070060528201,

411: 0, 1,

412: 0, 1,

413: 0, 1, 12227326696522585,

414: 0, 1,

415: 0, 1,

416: 0, 1, 12865948772698045,

417: 0, 1,

418: 0, 1,

419: 0, 1, 13532949699069781,

420: 0, 1,

421: 0, 1,

422: 0, 1, 14229372689107381,

423: 0, 1,

424: 0, 1,

425: 0, 1, 14956291412325901,

426: 0, 1,

427: 0, 1,

428: 0, 1, 15714810656334505,

429: 0, 1,

430: 0, 1,

431: 0, 1, 16506066998410705,

432: 0, 1,

433: 0, 1,

434: 0, 1, 17331229486668241,

435: 0, 1,

436: 0, 1,

437: 0, 1, 18191500330886641,

438: 0, 1,

439: 0, 1,

440: 0, 1, 19088115603070501,

441: 0, 1,

442: 0, 1,

443: 0, 1, 20022345947806525,

444: 0, 1,

445: 0, 1,

446: 0, 1, 20995497302486365,

447: 0, 1,

448: 0, 1,

449: 0, 1, 22008911627463301,

450: 0, 1,

451: 0, 1,

452: 0, 1, 23063967646210801,

453: 0, 1,

454: 0, 1,

455: 0, 1, 24162081595551001,

456: 0, 1,

457: 0, 1,

458: 0, 1, 25304707986021145,

459: 0, 1,

460: 0, 1,

461: 0, 1, 26493340372446025,

462: 0, 1,

463: 0, 1,

464: 0, 1, 27729512134784461,

465: 0, 1,

466: 0, 1,

467: 0, 1, 29014797269317861,

468: 0, 1,

469: 0, 1,

470: 0, 1, 30350811190248901,

471: 0, 1,

472: 0, 1,

473: 0, 1, 31739211541778365,

474: 0, 1,

475: 0, 1,

476: 0, 1, 33181699020728185,

477: 0, 1,

478: 0, 1,

479: 0, 1, 34680018209778721,

480: 0, 1,

481: 0, 1,

482: 0, 1, 36235958421388321,

483: 0, 1,

484: 0, 1,

485: 0, 1, 37851354552463201,

486: 0, 1,

487: 0, 1,

488: 0, 1, 39528087949845685,

489: 0, 1,

490: 0, 1,

491: 0, 1, 41268087286688845,

492: 0, 1,

493: 0, 1,

494: 0, 1, 43073329449785581,

495: 0, 1,

496: 0, 1,

497: 0, 1, 44945840437920181,

498: 0, 1,

499: 0, 1,

500: 0, 1, 46887696271310401,

501: 0, 1,

502: 0, 1,

503: 0, 1, 48901023912208105,

504: 0, 1,

505: 0, 1,

506: 0, 1, 50988002196726505,

507: 0, 1,

508: 0, 1,

509: 0, 1, 53150862777962041,

510: 0, 1,

511: 0, 1,

512: 0, 1, 55391891080478941,

513: 0, 1,

514: 0, 1,

515: 0, 1, 57713427266224501,

516: 0, 1,

517: 0, 1,

518: 0, 1, 60117867211943125,

519: 0, 1,

520: 0, 1,

521: 0, 1, 62607663498157165,

522: 0, 1,

523: 0, 1,

524: 0, 1, 65185326409782601,

525: 0, 1,

526: 0, 1,

527: 0, 1, 67853424948447601,

528: 0, 1,

529: 0, 1,

530: 0, 1, 70614587856582001,

531: 0, 1,

532: 0, 1,

533: 0, 1, 73471504653345745,

534: 0, 1,

535: 0, 1,

536: 0, 1, 76426926682464325,

537: 0, 1,

538: 0, 1,

539: 0, 1, 79483668172039261,

540: 0, 1,

541: 0, 1,

542: 0, 1, 82644607306401661,

543: 0, 1,

544: 0, 1,

545: 0, 1, 85912687310076901,

546: 0, 1,

547: 0, 1,

548: 0, 1, 89290917543928465,

549: 0, 1,

550: 0, 1,

551: 0, 1, 92782374613548985,

552: 0, 1,

553: 0, 1,

554: 0, 1, 96390203489966521,

555: 0, 1,

556: 0, 1,

557: 0, 1, 100117618642734121,

558: 0, 1,

559: 0, 1,

560: 0, 1, 103967905185470701,

561: 0, 1,

562: 0, 1,

563: 0, 1, 107944420033921285,

564: 0, 1,

565: 0, 1,

566: 0, 1, 112050593076604645,

567: 0, 1,

568: 0, 1,

569: 0, 1, 116289928358116381,

570: 0, 1,

571: 0, 1,

572: 0, 1, 120666005275155481,

573: 0, 1,

574: 0, 1,

575: 0, 1, 125182479785342401,

576: 0, 1,

577: 0, 1,

578: 0, 1, 129843085628896705,

579: 0, 1,

580: 0, 1,

581: 0, 1, 134651635563242305,

582: 0, 1,

583: 0, 1,

584: 0, 1, 139612022610608341,

585: 0, 1,

586: 0, 1,

587: 0, 1, 144728221318693741,

588: 0, 1,

589: 0, 1,

590: 0, 1, 150004289034463501,

591: 0, 1,

592: 0, 1,

593: 0, 1, 155444367191144725,

594: 0, 1,

595: 0, 1,

596: 0, 1, 161052682608490465,

597: 0, 1,

598: 0, 1,

599: 0, 1, 166833548806379401,

600: 0, 1,

601: 0, 1,

602: 0, 1, 172791367331819401,

603: 0, 1,

604: 0, 1,

605: 0, 1, 178930629099423001,

606: 0, 1,

607: 0, 1,

608: 0, 1, 185255915745422845,

609: 0, 1,

610: 0, 1,

611: 0, 1, 191771900995295125,

612: 0, 1,

613: 0, 1,

614: 0, 1, 198483352045059061,

615: 0, 1,

616: 0, 1,

617: 0, 1, 205395130956320461,

618: 0, 1,

619: 0, 1,

620: 0, 1, 212512196065127401,

621: 0, 1,

622: 0, 1,

623: 0, 1, 219839603404706065,

624: 0, 1,

625: 0, 1,

626: 0, 1, 227382508142144785,

627: 0, 1,

628: 0, 1,

629: 0, 1, 235146166029094321,

630: 0, 1,

631: 0, 1,

632: 0, 1, 243135934866552421,

633: 0, 1,

634: 0, 1,

635: 0, 1, 251357275983800701,

636: 0, 1,

637: 0, 1,

638: 0, 1, 259815755731561885,

639: 0, 1,

640: 0, 1,

641: 0, 1, 268517046989445445,

642: 0, 1,

643: 0, 1,

644: 0, 1, 277466930687749681,

645: 0, 1,

646: 0, 1,

647: 0, 1, 286671297343688281,

648: 0, 1,

649: 0, 1,

650: 0, 1, 296136148612109401,

651: 0, 1,

652: 0, 1,

653: 0, 1, 305867598850775305,

654: 0, 1,

655: 0, 1,

656: 0, 1, 315871876700270605,

657: 0, 1,

658: 0, 1,

659: 0, 1, 326155326678607141,

660: 0, 1,

661: 0, 1,

662: 0, 1, 336724410790593541,

663: 0, 1,

664: 0, 1,

665: 0, 1, 347585710152037501,

666: 0, 1,

667: 0, 1,

668: 0, 1, 358745926628848825,

669: 0, 1,

670: 0, 1,

671: 0, 1, 370211884491111265,

672: 0, 1,

673: 0, 1,

674: 0, 1, 381990532082191201,

675: 0, 1,

676: 0, 1,

677: 0, 1, 394088943502951201,

678: 0, 1,

679: 0, 1,

680: 0, 1, 406514320311136501,

681: 0, 1,

682: 0, 1,

683: 0, 1, 419273993236002445,

684: 0, 1,

685: 0, 1,

686: 0, 1, 432375423908250925,

687: 0, 1,

688: 0, 1,

689: 0, 1, 445826206605343861,

690: 0, 1,

691: 0, 1,

692: 0, 1, 459634070012261761,

693: 0, 1,

694: 0, 1,

695: 0, 1, 473806878997775401,

696: 0, 1,

697: 0, 1,

698: 0, 1, 488352636406298665,

699: 0, 1,

700: 0, 1,

701: 0, 1, 503279484865390585,

702: 0, 1,

703: 0, 1,

704: 0, 1, 518595708608974621,

705: 0, 1,

706: 0, 1,

707: 0, 1, 534309735316343221,

708: 0, 1,

709: 0, 1,

710: 0, 1, 550430137967015701,

711: 0, 1,

712: 0, 1,

713: 0, 1, 566965636711517485,

714: 0, 1,

715: 0, 1,

716: 0, 1, 583925100758148745,

717: 0, 1,

718: 0, 1,

719: 0, 1, 601317550275810481,

720: 0, 1,

721: 0, 1,

722: 0, 1, 619152158312956081,

723: 0, 1,

724: 0, 1,

725: 0, 1, 637438252732736401,

726: 0, 1,

727: 0, 1,

728: 0, 1, 656185318164406405,

729: 0, 1,

730: 0, 1,

731: 0, 1, 675402997971061405,

732: 0, 1,

733: 0, 1,

734: 0, 1, 695101096233770941,

735: 0, 1,

736: 0, 1,

737: 0, 1, 715289579752178341,

738: 0, 1,

739: 0, 1,

740: 0, 1, 735978580061634001,

741: 0, 1,

742: 0, 1,

743: 0, 1, 757178395466930425,

744: 0, 1,

745: 0, 1,

746: 0, 1, 778899493092707065,

747: 0, 1,

748: 0, 1,

749: 0, 1, 801152510950593001,

750: 0, 1,

751: 0, 1,

752: 0, 1, 823948260023155501,

753: 0, 1,

754: 0, 1,

755: 0, 1, 847297726364722501,

756: 0, 1,

757: 0, 1,

758: 0, 1, 871212073219147045,

759: 0, 1,

760: 0, 1,

761: 0, 1, 895702643154581725,

762: 0, 1,

763: 0, 1,

764: 0, 1, 920780960215331161,

765: 0, 1,

766: 0, 1,

767: 0, 1, 946458732090850561,

768: 0, 1,

769: 0, 1,

770: 0, 1, 972747852301958401,

771: 0, 1,

772: 0, 1,

773: 0, 1, 999660402404331265,

774: 0, 1,

775: 0, 1,

776: 0, 1, 1027208654209348885,

777: 0, 1,

778: 0, 1,

779: 0, 1, 1055405072022357421,

780: 0, 1,

781: 0, 1,

782: 0, 1, 1084262314898419021,

783: 0, 1,

784: 0, 1,

785: 0, 1, 1113793238915615701,

786: 0, 1,

787: 0, 1,

788: 0, 1, 1144010899465975585,

789: 0, 1,

790: 0, 1,

791: 0, 1, 1174928553564089545,

792: 0, 1,

793: 0, 1,

794: 0, 1, 1206559662173486281,

795: 0, 1,

796: 0, 1,

797: 0, 1, 1238917892550833881,

798: 0, 1,

799: 0, 1,

800: 0, 1, 1272017120608035901,

801: 0, 1,

802: 0, 1,

803: 0, 1, 1305871433292290005,

804: 0, 1,

805: 0, 1,

806: 0, 1, 1340495130984177205,

807: 0, 1,

808: 0, 1,

809: 0, 1, 1375902729913849741,

810: 0, 1,

811: 0, 1,

812: 0, 1, 1412108964595385641,

813: 0, 1,

814: 0, 1,

815: 0, 1, 1449128790279378001,

816: 0, 1,

817: 0, 1,

818: 0, 1, 1486977385423827025,

819: 0, 1,

820: 0, 1,

821: 0, 1, 1525670154183402865,

822: 0, 1,

823: 0, 1, 197427385,

824: 0, 1, 1565222728917147301,

825: 0, 1,

826: 0, 1,

827: 0, 1, 1605650972714682301,

828: 0, 1,

829: 0, 1,

830: 0, 1, 1646970981940993501,

831: 0, 1,

832: 0, 1,

833: 0, 1, 1689199088799856645,

834: 0, 1,

835: 0, 1,

836: 0, 1, 1732351863915975025,

837: 0, 1,

838: 0, 1,

839: 0, 1, 1776446118935895961,

840: 0, 1,

841: 0, 1,

842: 0, 1, 1821498909147774361,

843: 0, 1,

844: 0, 1,

845: 0, 1, 1867527536120051401,

846: 0, 1,

847: 0, 1,

848: 0, 1, 1914549550359116365,

849: 0, 1,

850: 0, 1,

851: 0, 1, 1962582753986019685,

852: 0, 1,

853: 0, 1,

854: 0, 1, 2011645203432305221,

855: 0, 1,

856: 0, 1,

857: 0, 1, 2061755212155029821,

858: 0, 1,

859: 0, 1,

860: 0, 1, 2112931353371038201,

861: 0, 1,

862: 0, 1,

863: 0, 1, 2165192462810561185,

864: 0, 1,

865: 0, 1,

866: 0, 1, 2218557641490205345,

867: 0, 1,

868: 0, 1,

869: 0, 1, 2273046258505402081,

870: 0, 1,

871: 0, 1,

872: 0, 1, 2328677953842384181,

873: 0, 1,

874: 0, 1,

875: 0, 1, 2385472641209757901,

876: 0, 1,

877: 0, 1,

878: 0, 1, 2443450510889738605,

879: 0, 1,

880: 0, 1,

881: 0, 1, 2502632032609118005,

882: 0, 1,

883: 0, 1,

884: 0, 1, 2563037958430031041,

885: 0, 1,

886: 0, 1,

887: 0, 1, 2624689325660590441,

888: 0, 1,

889: 0, 1,

890: 0, 1, 2687607459785457001,

891: 0, 1,

892: 0, 1,

893: 0, 1, 2751813977416413625,

894: 0, 1,

895: 0, 1,

896: 0, 1, 2817330789263011165,

897: 0, 1,

898: 0, 1,

899: 0, 1, 2884180103123354101,

900: 0, 1,

901: 0, 1,

902: 0, 1, 2952384426895094101,

903: 0, 1,

904: 0, 1,

905: 0, 1, 3021966571606699501,

906: 0, 1,

907: 0, 1,

908: 0, 1, 3092949654469068745,

909: 0, 1,

910: 0, 1,

911: 0, 1, 3165357101947555825,

912: 0, 1,

913: 0, 1,

914: 0, 1, 3239212652854475761,

915: 0, 1,

916: 0, 1,

917: 0, 1, 3314540361462158161,

918: 0, 1,

919: 0, 1,

920: 0, 1, 3391364600636616901,

921: 0, 1,

922: 0, 1,

923: 0, 1, 3469710064991903965,

924: 0, 1,

925: 0, 1,

926: 0, 1, 3549601774065215485,

927: 0, 1,

928: 0, 1,

929: 0, 1, 3631065075512818021,

930: 0, 1,

931: 0, 1,

932: 0, 1, 3714125648326863121,

933: 0, 1,

934: 0, 1,

935: 0, 1, 3798809506073158201,

936: 0, 1,

937: 0, 1,

938: 0, 1, 3885143000149961785,

939: 0, 1,

940: 0, 1,

941: 0, 1, 3973152823067871145,

942: 0, 1,

943: 0, 1,

944: 0, 1, 4062866011750870381,

945: 0, 1,

946: 0, 1,

947: 0, 1, 4154309950858606981,

948: 0, 1,

949: 0, 1,

950: 0, 1, 4247512376129964901,

951: 0, 1,

952: 0, 1,

953: 0, 1, 4342501377748002205,

954: 0, 1,

955: 0, 1,

956: 0, 1, 4439305403726321305,

957: 0, 1,

958: 0, 1,

959: 0, 1, 4537953263316939841,

960: 0, 1,

961: 0, 1,

962: 0, 1, 4638474130439730241,

963: 0, 1,

964: 0, 1,

965: 0, 1, 4740897547133496001,

966: 0, 1,

967: 0, 1,

968: 0, 1, 4845253427028752725,

969: 0, 1,

970: 0, 1,

971: 0, 1, 4951572058842281965,

972: 0, 1,

973: 0, 1,

974: 0, 1, 5059884109893525901,

975: 0, 1,

976: 0, 1,

977: 0, 1, 5170220629642890901,

978: 0, 1,

979: 0, 1,

980: 0, 1, 5282613053252028001,

981: 0, 1,

982: 0, 1,

983: 0, 1, 5397093205166158345,

984: 0, 1,

985: 0, 1,

986: 0, 1, 5513693302718511625,

987: 0, 1,

988: 0, 1,

989: 0, 1, 5632445959756945561,

990: 0, 1,

991: 0, 1,

992: 0, 1, 5753384190292814461,

993: 0, 1,

994: 0, 1,

995: 0, 1, 5876541412172154901,

996: 0, 1,

997: 0, 1,

998: 0, 1, 6001951450769256565,

999: 0, 1,

1000: 0, 1,

2021年5月16日日曜日

210516

Ruby

Sum_{n = 1..24} n^2 = 70^2(4)

https://www.youtube.com/watch?v=q6L06pyt9CA

この動画に出てくる内容について出力してみた。

# polygonal number
def p_k(k, n)
  n * ((k - 2) * n - k + 4) / 2
end

# Sum_{j=1..n} p_k(k, j)
def q_k(k, n)
  (1..n).inject(0){|s, i| s + p_k(k, i)}
end

def f(k, m, n)
  p k
  p [m, q_k(k, m)]
  p [n, p_k(k, n)]
end

f(4, 24, 70)
f(8, 10, 19)
f(8, 18, 45)
f(31265, 259, 2407)

出力結果

[24, 4900]

[70, 4900]

[10, 1045]

[19, 1045]

[18, 5985]

[45, 5985]

31265

[259, 90525801730]

[2407, 90525801730]

2021年5月1日土曜日

210501

Ruby

(1/e) * Sum_{a_1>=1, a_2>=1, ... , a_n>=1} a_1 * a_2 * ... * a_k / (a_1 + a_2 + ... + a_n)!

O. Furdui, Limits, Series and Fractional Part Integrals

の３章のOpen Problemの一つに

Sum_{a_1>=1, a_2>=1, ... , a_n>=1} a_1 * a_2 * ... * a_n / (a_1 + a_2 + ... + a_n)!

が有理数 * e と書けるかいう問題があるが、

より一般的に、1 <= k <= n のとき、

Sum_{a_1>=1, a_2>=1, ... , a_n>=1} a_1 * a_2 * ... * a_k / (a_1 + a_2 + ... + a_n)!

が有理数 * e と書ける。

この有理数すなわち

(1/e) * Sum_{a_1>=1, a_2>=1, ... , a_n>=1} a_1 * a_2 * ... * a_k / (a_1 + a_2 + ... + a_n)!

を出力してみた。

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end

def A(k, n)
  (0..k - 1).inject(0){|s, i| s + ncr(k - 1, i) / f(i + n).to_r}
end

n = 6
(1..n).each{|i|
  p [i, (1..i).map{|j| A(j, i)}]
}

出力結果

[1, [(1/1)]]

[2, [(1/2), (2/3)]]

[3, [(1/6), (5/24), (31/120)]]

[4, [(1/24), (1/20), (43/720), (179/2520)]]

[5, [(1/120), (7/720), (19/1680), (529/40320), (787/51840)]]

[6, [(1/720), (1/630), (73/40320), (187/90720), (8501/3628800), (6631/2494800)]]

2021年4月3日土曜日

210403

Ruby

Number of tilings of a <n,k,n> hexagon

一般化された二項係数を利用し計算してみた。

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end

# generalized binomial coefficient (n,r)_m
def g_ncr_m(n, r, m = 1)
  # to_rとto_iは念のため
  (1..r).inject(1){|s, i| s * ncr(n - i + m, m) / ncr(i - 1 + m, m).to_r}.to_i
end

n = 10
(0..n).each{|i|
  p [i, (0..5).map{|j| g_ncr_m(i + j, i, i)}]
}

出力結果

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

[1, [1, 2, 3, 4, 5, 6]]

[2, [1, 6, 20, 50, 105, 196]]

[3, [1, 20, 175, 980, 4116, 14112]]

[4, [1, 70, 1764, 24696, 232848, 1646568]]

[5, [1, 252, 19404, 731808, 16818516, 267227532]]

[6, [1, 924, 226512, 24293412, 1447482465, 55197331332]]

[7, [1, 3432, 2760615, 877262100, 142174944340, 13710834632352]]

[8, [1, 12870, 34763300, 33803832920, 15484613937936, 3940599631842016]]

[9, [1, 48620, 449141836, 1371597504992, 1832516612010448, 1274867710694037824]]

[10, [1, 184756, 5924217936, 58043512597616, 232187445047217296, 454924095627897307776]]

2021年3月6日土曜日

210306

Ruby

チェビシェフ多項式の明示的な式を利用した計算

T_n(n+k), U_n(n+k)の値を、

明示的な式を利用し計算してみた。

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end

# n >= 0, m = 1 or 2
def chebyshev(n, m, k)
  if m == 2
    return (0..n).inject(0){|s, i| s + (2 * k - 2) ** i * ncr(n + 1 + i, 2 * i + 1)}
  elsif m == 1
    if n == 0
      return 1
    else
      return (n * (0..n).inject(0){|s, i| s + (2 * k - 2) ** i * ncr(n + i, 2 * i) / (n + i.to_r)}).to_i
    end
  end
end

K = 5
N = 9
(0..K).each{|k|
  p [k, (0..N).map{|i| chebyshev(i, 1, i + k)}]
}
(0..K).each{|k|
  p [k, (0..N).map{|i| chebyshev(i, 2, i + k)}]
}

出力結果

[0, [1, 1, 7, 99, 1921, 47525, 1431431, 50843527, 2081028097, 96450076809]]

[1, [1, 2, 17, 244, 4801, 120126, 3650401, 130576328, 5374978561, 250283080090]]

[2, [1, 3, 31, 485, 10081, 262087, 8193151, 299537289, 12545596801, 592479412811]]

[3, [1, 4, 49, 846, 18817, 514088, 16692641, 628855930, 26986755841, 1300371936876]]

[4, [1, 5, 71, 1351, 32257, 930249, 31521799, 1229215691, 54276558337, 2678768828749]]

[5, [1, 6, 97, 2024, 51841, 1580050, 55989361, 2265463212, 103182433537, 5228741809214]]

[0, [1, 2, 15, 204, 3905, 96030, 2883167, 102213944, 4178507265, 193501094490]]

[1, [1, 4, 35, 496, 9701, 241956, 7338631, 262184896, 10783446409, 501827040100]]

[2, [1, 6, 63, 980, 20305, 526890, 16451071, 600940872, 25154396001, 1187422368110]]

[3, [1, 8, 99, 1704, 37829, 1032240, 33489287, 1260879920, 54085723209, 2605282707576]]

[4, [1, 10, 143, 2716, 64769, 1866294, 63202399, 2463542488, 108742564225, 5365498355074]]

[5, [1, 12, 195, 4064, 104005, 3168060, 112211527, 4538833824, 206671502025, 10470873504140]]

2021年2月23日火曜日

210223

PARI

Number of perfect matchings in the graph C_{2n} X C_k

kが3以上のとき、計算してみた。

a341533(n, k) = round(sqrt(prod(a=1, n,   prod(b=1, k,   4*sin((2*a-1)*Pi/(2*n))^2+4*sin((2*b-1)*Pi/k)^2))));
a341738(n, k) = round(sqrt(prod(a=1, n,   prod(b=1, k-1, 4*sin((2*a-1)*Pi/(2*n))^2+4*sin(2*b*Pi/k)^2))));
a341739(n, k) = round(     prod(a=1, n-1, prod(b=1, k,   4*sin(a*Pi/n)^2          +4*sin((2*b-1)*Pi/(2*k))^2)));
T(n, k) = a341533(n, k)/2+a341738(n, k)+2*((k+1)%2)*a341739(n, ceil(k/2));

K=7;
for(n=2, 10, for(k=3, K, print1(T(n, k), ", ")); print)

出力結果

50, 272, 722, 3108, 10082,

224, 3108, 9922, 90176, 401998,

1058, 39952, 155682, 3113860, 19681538,

5054, 537636, 2540032, 114557000, 1034315998,

24200, 7379216, 41934482, 4357599552, 55820091938,

115934, 102144036, 694861522, 169094614280, 3044533460992,

555458, 1418981392, 11527389122, 6640849944580, 166779871224962,

2661344, 19742287908, 191304901282, 262671237617216, 9152970837103102,

12751250, 274848860432, 3175220160032, 10433179552323108, 502711247500143362,

2021年1月17日日曜日

210117

Python

Number of spanning trees of odd Aztec diamond OD_n

Graphillion を使って計算してみた。

from graphillion import GraphSet

def make_OD_nk(n, k):
    n += 1
    k += 1
    grids = []
    s = k * n
    for i in range(1, k * n, k):
        for j in range(1, k):
            a, b = i + j - 1, i + j
            c = s + a
            if i > 1:
                grids.extend([(c - k, a), (c - k, b)])
            if i < k * (n - 1) + 1:
                grids.extend([(c, a), (c, b)])
    return grids

def A340425(n, k):
    universe = make_OD_nk(n, k)
    GraphSet.set_universe(universe)
    spanning_trees = GraphSet.trees(is_spanning=True)
    return spanning_trees.len()

print([A340425(n, n) for n in range(1, 6)])

出力結果

[1, 192, 4542720, 12116689944576, 3544863978266468352000]

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