181020

Ruby


Theta series of D_k lattice

k <= 96 のとき、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

def B(k, n)
  a = A([[2, 5 * k], [1, -2 * k], [4, -2 * k]], 2 * n)
  (0..n).map{|i| a[2 * i]}
end

n = 20
(1..96).each{|i| p [i, B(i, n)]}

出力結果
[1, [1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0]]
[2, [1, 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0, 4, 8, 4, 0, 8]]
[3, [1, 12, 6, 24, 12, 24, 8, 48, 6, 36, 24, 24, 24, 72, 0, 48, 12, 48, 30, 72, 24]]
[4, [1, 24, 24, 96, 24, 144, 96, 192, 24, 312, 144, 288, 96, 336, 192, 576, 24, 432, 312, 480, 144]]
[5, [1, 40, 90, 240, 200, 560, 400, 800, 730, 1240, 752, 1840, 1200, 2000, 1600, 2720, 1480, 3680, 2250, 3280, 2800]]
[6, [1, 60, 252, 544, 1020, 1560, 2080, 3264, 4092, 4380, 6552, 8160, 8224, 10200, 12480, 14144, 16380, 17400, 18396, 24480, 26520]]
[7, [1, 84, 574, 1288, 3444, 4424, 9240, 11088, 18494, 19740, 34440, 31304, 52808, 52248, 74048, 71120, 110964, 94864, 145222, 132888, 181384]]
[8, [1, 112, 1136, 3136, 9328, 14112, 31808, 38528, 74864, 84784, 143136, 149184, 261184, 246176, 390784, 395136, 599152, 550368, 859952, 768320, 1175328]]
[9, [1, 144, 2034, 7392, 22608, 44640, 93984, 141120, 262386, 343536, 557280, 703584, 1068384, 1234080, 1851264, 2057280, 2884176, 3231936, 4428242, 4639392, 6455520]]
[10, [1, 180, 3380, 16320, 52020, 129064, 262080, 489600, 840500, 1330420, 2050344, 2986560, 4194240, 5878440, 7862400, 10216320, 13415220, 17163880, 21250420, 26585280, 32826664]]
[11, [1, 220, 5302, 33528, 116380, 339064, 719400, 1538416, 2624182, 4763220, 7217144, 11676280, 16317048, 25022184, 32503680, 47430768, 59759260, 82927856, 101189550, 137904360, 161928184]]
[12, [1, 264, 7944, 64416, 253704, 825264, 1938336, 4437312, 8118024, 15653352, 24832944, 42517728, 61903776, 98021616, 133522752, 201364416, 259776264, 374842512, 471023592, 653690400, 793078704]]
[13, [1, 312, 11466, 116688, 535704, 1899664, 5043376, 12066912, 24443978, 48210760, 83323344, 145532816, 227110416, 363737712, 531517376, 800145632, 1104376728, 1594656544, 2114015514, 2931991920, 3779027408]]
[14, [1, 364, 16044, 200928, 1089452, 4196920, 12547808, 31553344, 70439852, 142487436, 268594872, 476105504, 800698080, 1296257144, 2022013760, 3056208064, 4507001772, 6481076056, 9123064524, 12637846368, 17190762680]]
[15, [1, 420, 21870, 331240, 2128260, 8972712, 29822520, 80027280, 193511790, 409172940, 825736296, 1509537960, 2698852520, 4472616120, 7347537600, 11335886864, 17520853380, 25553816400, 37658603350, 52706867640, 74655559272]]
[16, [1, 480, 29152, 525952, 3994080, 18626112, 67978880, 197120256, 509145568, 1143441760, 2428524096, 4658843520, 8705492608, 15000919872, 25593401600, 40864033536, 65187410400, 98098623936, 148724365664, 213678303360, 310833534528]]
[17, [1, 544, 38114, 808384, 7213984, 37569728, 149405248, 470966912, 1291650786, 3101265120, 6883061184, 13967503552, 27036220096, 48877735232, 85898222336, 142979339392, 233808578464, 365627715968, 565633846434, 841764350272, 1246798917056]]
[18, [1, 612, 48996, 1207680, 12573540, 73617480, 318102912, 1090632960, 3176573796, 8148505828, 18918517320, 40578145920, 81375600000, 154398101832, 279331331328, 484986858240, 812932819812, 1320330798792, 2085523894756, 3214498469760, 4845373383240]]
[19, [1, 684, 62054, 1759704, 21210156, 140116184, 658369608, 2449182384, 7597766246, 20748447108, 50601708504, 114223115480, 238352317656, 472600643976, 883678139136, 1594639240368, 2749536787500, 4620722631088, 7482364853310, 11894205734280, 18321074279384]]
[20, [1, 760, 77560, 2508000, 34729720, 259114704, 1327461600, 5341699520, 17701924600, 51294999960, 131880275664, 312126610080, 680432137440, 1404010658960, 2724593155520, 5089580732736, 9062110326520, 15699349246320, 26159307313560, 42719885410400, 67517852274384]]
[21, [1, 840, 95802, 3504816, 55350120, 465944304, 2610853840, 11337146784, 40208417466, 123411903160, 335056780464, 830096613360, 1893781702512, 4058779021584, 8190877926720, 15805879161632, 29122237029480, 51905458442208, 89163973663338, 149313692762640, 242598110651568]]
[22, [1, 924, 117084, 4812192, 86075484, 815677016, 5012882336, 23460347328, 89096422492, 289559673084, 830202403800, 2153312518240, 5140361517984, 11444537517336, 24015334687168, 47873378766400, 91281903219804, 167369436673208, 296418599695548, 509007239863584, 850134257934808]]
[23, [1, 1012, 141726, 6503112, 130905236, 1392036360, 9401213016, 47412641232, 192747766686, 663612939132, 2007831887432, 5456890474248, 13617909938952, 31527898462552, 68719323996224, 141662701280848, 279248601010324, 527242669347856, 961796734667878, 1695182770403416, 2907618520941640]]
[24, [1, 1104, 170064, 8662720, 195082320, 2319457632, 17231109824, 93703589760, 407488018512, 1487286966928, 4744779429216, 13523760003648, 35250721087168, 84944161233120, 192127452262272, 409984511707776, 834658207381584, 1624411151609760, 3049176879087248, 5521413521657280, 9716572902143328]]
[25, [1, 1200, 202450, 11389600, 285385200, 3779134240, 30885301600, 181284196800, 842726826450, 3262594952400, 10968671291040, 32802221530400, 89267475949600, 223991253535200, 525491835433600, 1161295395893440, 2440508069871600, 4898380398568000, 9456583413620850, 17601674014456800, 31764761129879200]]
[26, [1, 1300, 239252, 14797120, 410468500, 6030043240, 54177864000, 343594659200, 1706685733012, 7010460268500, 24832297608040, 77924078814400, 221394909419840, 578472382307304, 1407681394953600, 3221568346322560, 6988918697572500, 14466438251512104, 28723680851525972, 54955933066382400, 101702425588365160]]
[27, [1, 1404, 280854, 19014840, 581257404, 9436118328, 93086496360, 638406025200, 3387901252374, 14765641285044, 55108831380408, 181434042582840, 538265644820280, 1464212332875816, 3696677780664960, 8759016189382896, 19620550996489404, 41872709357187696, 85529698383793614, 168166018677618216, 319216925216931768]]
[28, [1, 1512, 327656, 24189984, 811401192, 14500933104, 156797510688, 1163495873088, 6597606440936, 30505974273096, 119983597365744, 414352982942304, 1283861585339424, 3635198568784944, 9523977599887936, 23358459198626496, 54039921152557032, 118876818364991568, 249860594326120008, 504728808620994720, 982979367330327024]]
[29, [1, 1624, 380074, 30488976, 1117791544, 21911472464, 259172461296, 2081117152992, 12613606876714, 61865177248552, 256462892399504, 928859670458448, 3006249397711056, 8858986222461872, 24088488255932096, 61146089973555552, 146117613009432632, 331281589561600160, 716578329768711354, 1486998916193387184, 2971591747809448848]]
[30, [1, 1740, 438540, 38099040, 1521151500, 32592806520, 420775407200, 3655476494400, 23690038522380, 123233778584940, 538518903906552, 2045337982427040, 6914885182908000, 21207036616238520, 59847918853661760, 157231420280219584, 388094025605841420, 906869192637613080, 2018715060481365420, 4303383637492581600, 8824294692839138040]]
[31, [1, 1860, 503502, 47229864, 2046701220, 47775740520, 671636175480, 6309132101904, 43733123309262, 241275541236780, 1111479953165352, 4426891212365160, 15633379938039144, 49899492947343864, 146147632445771200, 397406328481552528, 1013148957319630500, 2440161908107807952, 5589676687772302966, 12241554552984167160, 25755553416637335080]]
[32, [1, 1984, 575424, 58115328, 2724906944, 69079796864, 1054968628480, 10706472186368, 79395777333184, 464573878394560, 2256126097001600, 9423431444080384, 34759556728866048, 115474354635222656, 350982387781513728, 987881573708613120, 2601123015997179840, 6457558989366956928, 15221194949728018112, 34248312026351479040, 73928444877139381376]]
[33, [1, 2112, 654786, 71015296, 3592319808, 98614192512, 1632114867328, 17875388306688, 141820480610754, 880219339775424, 4506220683398016, 19739221443625344, 76047426405791104, 262953752848837248, 829405814646351360, 2416450963949279488, 6571105534323612480, 16815939617686463232, 40785056696578571586, 94285427570798699136, 208805653293610437504]]
[34, [1, 2244, 742084, 86217472, 4692511428, 139099807880, 2489047659264, 29382088665600, 249369222884036, 1641869055038916, 8860642360669320, 40707711590868224, 163797472570290944, 589511537253470856, 1929572788503833088, 5819255124075374080, 16342941364998777540, 43111163441224005000, 107589173820434997188, 255543757747733874432, 580614999165620921480]]
[35, [1, 2380, 837830, 104039320, 6077113420, 194015502744, 3744835385480, 47578912132400, 431833448394950, 3016461775970020, 17160396820563864, 82689238761879960, 347497817081977240, 1301746850099643080, 4421575623848670720, 13803058326450694448, 40035461652998329420, 108862061348826948400, 279550204291077467230, 682187522811751672520, 1590223759684822652824]]
[36, [1, 2520, 942552, 124830048, 7806968280, 267772521744, 5562557800800, 75951250082496, 736831099167192, 5460799823164920, 32748641010875664, 165514090360142880, 726465611491844448, 2832525877217028048, 9984123911261833920, 32262229857439895616, 96644313054689681880, 270878531986080745776, 715761543475698773496, 1794541390237379155680, 4291860930734291909904]]
[37, [1, 2664, 1056794, 148972656, 9953399304, 365921147824, 8163258314256, 119597548195872, 1239391811991194, 9745233570636952, 61609295691064944, 326597524550490288, 1497196642542832176, 6075984718042868304, 22225028881973326912, 74337473128386333600, 229988493902004444936, 664457654598844109152, 1806655171676194402314, 4653698168319059461968, 11419085895334809136752]]
[38, [1, 2812, 1181116, 176886048, 12599607484, 495394207384, 11843630888736, 185886180236992, 2056130097966268, 17150568272891100, 114303284464647704, 635553953336625376, 3043086948436643616, 12853691613921890904, 48791664731735114432, 168923174867580925248, 539765733887226387644, 1607416101409060228472, 4497294755894357458332, 11901773369017113604768, 29963055119264915340312]]
[39, [1, 2964, 1316094, 209027208, 15842203572, 664792507080, 16998268681368, 285345150337104, 3365941445206782, 29777240693878236, 209215435001112072, 1220155819685626824, 6102142531936618440, 26826945551844218520, 105677341468592172864, 378706598865712166608, 1249787083449735359412, 3836378587257542276496, 11044821525347480286598, 30030157748537873897880, 77566119196460997440520]]
[40, [1, 3120, 1462320, 245893440, 19792883760, 884717790112, 24147448985920, 432855539095680, 5439867081224240, 51024388607161840, 377925692648758176, 2311828718036596160, 12076325113647462720, 55258802243177832480, 225894020902792394880, 837922349252103601536, 2855961811305420582960, 9036524878000691192160, 26770192778468917234160, 74780870224218969913920, 198172571789841855437728]]
[41, [1, 3280, 1620402, 288024672, 24580257680, 1168159340512, 33971596798880, 649237878131776, 8683708343211698, 86322811423907760, 673977940881914976, 4324346553745956320, 23594791016741569248, 112373740165845667616, 476716600185913087360, 1830368645195764804928, 6443191629889162086800, 21014288031859432240064, 64058450518363788497682, 183847011771941652626720, 499858750759811796384864]]
[42, [1, 3444, 1790964, 336005824, 30351837684, 1530940938408, 47353759443648, 963342807437952, 13698191148446196, 144242851393605300, 1187013596538690984, 7988222608927186752, 45526484181023926976, 225683472479132529960, 993544787069503165056, 3948627822151732074880, 14355605005513083277812, 48261418497124590978024, 151381566173619091602740, 446369593591372088412096, 1245153274370742690277800]]
[43, [1, 3612, 1974646, 390469240, 37276198620, 1992235478520, 65431639315112, 1414784111195760, 21363058099953526, 238149668458074388, 2065271888000259960, 14577365408975111160, 86778165276794876920, 447752053048303878504, 2045589436006114852224, 8415088149377413297520, 31597043700929688644316, 109493876778527685657584, 353405938952937766733806, 1070623418671287055194984, 3064098127861622944029560]]
[44, [1, 3784, 2172104, 452097184, 45545317576, 2575155209264, 89660973247648, 2057484301193024, 32953489081394376, 388649134382308008, 3550983885332760624, 26286715322506463200, 163448683981013707936, 877815757629273245552, 4161780665859803103040, 17721547148427132981696, 68723006258858639123656, 245476497087950706480144, 815277009565266028320936, 2537517139640843807601440, 7450972920603097999394864]]
[45, [1, 3960, 2384010, 521624400, 55377103320, 3307426231824, 121892317700400, 2964241181293920, 50299827862307850, 627163962734020360, 6035410491750694224, 46854014968659908880, 304297378385575293840, 1701047831240174822640, 8369306397107313831360, 36888834239882740396512, 147743644093286996957400, 543977021439433920875040, 1859033820761919044705370, 5944722654363693006155760, 17909106681549810556695888]]
[46, [1, 4140, 2611052, 599840736, 67018125420, 4222156618424, 164463601475040, 4232569247463744, 76004850661311596, 1001109838673418060, 10143480948171060024, 82571796820419222560, 560115286465036588512, 3259052873842515520248, 16640395332885527444800, 75919437264083991860928, 314037149331078085727340, 1191836115047264823513304, 4191165803741335074725004, 13769539726145045949802080, 42559856748226014373309752]]
[47, [1, 4324, 2853934, 687593832, 80746553284, 5358708267816, 220311144761016, 5992123438556048, 113736905985171502, 1581315551638626444, 16862566236863499240, 143916122317077701160, 1019604050911451615784, 6175012304042413993784, 32719823078748978144448, 154520158805605372057872, 660128103579641521215428, 2582424058096501396180560, 9344529589056656406012310, 31541489566055534873934904, 100023323889942538865069544]]
[48, [1, 4512, 3113376, 785791872, 96875315616, 6763683414720, 293102217821056, 8414075863671552, 168622365198254496, 2472562531531980832, 27736395024393085632, 248139579890464622208, 1835986185002939939712, 11573413786349169929664, 63640852572427508785920, 311097167722066085728512, 1372634797639492206592416, 5535010786414722734528832, 20609142765989970203160608, 71470177750397835003888000, 232531699106546664748589760]]
[49, [1, 4704, 3390114, 895406400, 115755491040, 8492037549632, 387392627620544, 11722890040874880, 247767164972339362, 3828424156205839008, 45154189480417663296, 423356084326686995008, 3271121112943667769408, 21461709941573992929216, 122472660390060456632576, 619706321449560969104768, 2823979399419216267131616, 11737878719907676947436672, 44972048668391014172152290, 160231480848980530771422144, 534863840491664871588174144]]
[50, [1, 4900, 3684900, 1017475200, 137779940900, 10608331387080, 508813279363200, 16211023387641600, 360945048625380900, 5871984218650395300, 72777694962263662280, 714910303873915900800, 5767878361495107715200, 39386275590645554758600, 233247498388449049388800, 1221661417281854603505920, 5749681423147539513982500, 24634122723827333941603400, 97118478236074066740808100, 355506356928628484760899200, 1217533065913380509499410120]]
[51, [1, 5100, 3998502, 1153105240, 163387195500, 13188135445080, 664290165581000, 22257188865735600, 521499719190179622, 8924532432265106500, 116166126969336296280, 1195211729173952947800, 10067656683832769582680, 71548211019296692710408, 439706216301392062982400, 2383873245883380529354160, 11587610945394460751112300, 51174435909416999296832688, 207601157641922638919028862, 780756675904183361598318600, 2743386985289268437091191640]]
[52, [1, 5304, 4331704, 1303475680, 193065605304, 16319601767248, 862302791665120, 30348922637092800, 747519847870453944, 13445004171207696984, 183682764146804624208, 1978773353590745048480, 17399315857793454638560, 128682265229505007371024, 820665015623406722525120, 4605445416232558058471232, 23120669480397861067643064, 105250933550818269152815344, 439354340397428980578162264, 1697623660229051142995133792, 6119991092311362063535770448]]
[53, [1, 5512, 4685306, 1469840944, 227357768872, 20105218332784, 1113186657215568, 41110338355629728, 1063360138194067066, 20083790276552219256, 287799059912477065264, 3244997658407131836784, 29779980555917284035824, 229189666553906219004816, 1516750028313527617393472, 8810527185645786672919584, 45682348251115860769246632, 214358532054417196218274528, 920751363346351916460411306, 3655186949954152577015389328, 13519388354490357354705463344]]
[54, [1, 5724, 5060124, 1653533856, 266865249564, 24663762759384, 1429486081204896, 55336102527840192, 1501598892693637404, 29755637702370479292, 446955414869923659864, 5272393666964582906208, 50489456563598746221216, 404311428686749230067608, 2776459590758215646884800, 16693853037913375742127168, 89396383038780568363350300, 432393630325638785907707448, 1911146097905933862282977916, 7794747225818563563414593568, 29579285203322402284706215512]]
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[91, [1, 16380, 42762902, 42663005112, 21764086360380, 6588396034605112, 1295909524667819880, 176080729397800786416, 17274175663085960158358, 1265427277090139696702260, 71091953734387332789105336, 3131104731257966337130084920, 110160087693924640469871811128, 3147771210094457910921401464104, 74163720922578781851414781476480, 1461196770930868770227527023739632, 24398426041106196000436535908496700, 349705217987520275438076190790350960, 4355394877255199989803540799673904718, 47681622265525550538204656325599456040, 463828457332260239329234885481560837816]]
[92, [1, 16744, 44706664, 45639388704, 23836575476584, 7391571770787568, 1490145781154223648, 207638337279435591232, 20901431808171087571816, 1571920437007982758524360, 90707281328774486265619184, 4105187088601110691768723552, 148462980023921803082302395936, 4361611457533988964079224000816, 105659992890831950804193409490496, 2140190615408047859753911608989376, 36728145591768309861740535542000488, 540782942355083020576545394774484816, 6914381754472605224040456863513950152, 77651423226335898644627458237465086624, 774212967726450701238590040698959487728]]
[93, [1, 17112, 46715946, 48786825360, 26079724388280, 8281815757880400, 1710743911564592112, 244382795792376488160, 25233787729158322345386, 1947636503830224752663208, 115398821925171608651997840, 5364862539565834368026940240, 199368684901271570955991415760, 6019983461662891400081968924848, 149900287024387022347391176180416, 3120682748882136405494357678556000, 55028221722529188825156651010235064, 832158636462134100337009000474685088, 10921236909212115282144087054252739386, 125801863120491528682452883033289725872, 1285474010726783262427662725149974575760]]
[94, [1, 17484, 48792204, 52113090144, 28505438891916, 9267418636559224, 1960917301475064928, 287091398223682749504, 30397579296803754161036, 2407105246343608381340908, 146395367791834060569078264, 6988869161480762101584052640, 266793822350790844792230835296, 8277239967221050032513757944504, 211790299553826541552433721616448, 4530432646846117469233313987675072, 82065460481051240691301591484171148, 1274352622729749638033978983590176536, 17164036390919984515254346771983096108, 202767617331836233977248502461871971040, 2123238376413291560845370754090521070584]]
[95, [1, 17860, 50936910, 55626220200, 31126309002020, 10357353717781864, 2244227657088347000, 336646918517988520720, 36539902637249730870350, 2967679941264725288031660, 185203108063366873037122344, 9076307982221197940766437480, 355802297161951565511600449640, 11338459650196245218411426852600, 298029500465521067020697392187840, 6548792182670474006057805059914384, 121832235729401926708138102115015460, 1942276010496907268952311182625936080, 26843067932361648971465425310531142390, 325175811603342569798980858407180658680, 3488986245373935202597276004650802119976]]
[96, [1, 18240, 53151552, 59334520576, 33955640051520, 11561324607022464, 2564618170533277440, 394050467550457419264, 43831724698408040859456, 3650043138612603399351360, 233663830844106046138230144, 11751569440230400021168968960, 472921360286535475153166266112, 15475194992617219000176432302976, 417733128820716408997571835778560, 9426531011128539893070734331875840, 180064687104473451358115250869962560, 2946503703724638803275689931394718336, 41777796122601853387625086803402058816, 518895417590322600275705085386080623360, 5704228691829759217304970570342609724800]]

181015(3)

Ruby


A320466, A320470, A240026 とA240027

n が50以下のとき、出力してみた。

@ary1 = [1]
@ary2 = [1]
@ary3 = [1]
@ary4 = [1]

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end

# n > 0
def f(n)
  cnt1, cnt2, cnt3, cnt4 = 0, 0, 0, 0
  partition(n, 1, n).each{|ary|
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    ary1 = ary0.sort
    if ary1 == ary0
      cnt1 += 1
      cnt2 += 1 if ary0.uniq == ary0
    end
    if ary1 == ary0.reverse
      cnt3 += 1
      cnt4 += 1 if ary0.uniq == ary0
    end
  }
  @ary1 << cnt1
  @ary2 << cnt2
  @ary3 << cnt3
  @ary4 << cnt4
end

n = 50
(1..n).each{|i| f(i)}
puts "A320466"
p @ary1
puts "A320470"
p @ary2
puts "A240026"
p @ary3
puts "A240027"
p @ary4

出力結果
A320466
[1, 1, 2, 3, 4, 5, 7, 7, 9, 12, 12, 13, 18, 17, 21, 25, 24, 27, 34, 33, 38, 44, 43, 47, 58, 56, 62, 70, 70, 78, 90, 84, 96, 109, 108, 118, 132, 127, 140, 158, 158, 167, 189, 185, 204, 221, 218, 236, 260, 261, 282]
A320470
[1, 1, 2, 2, 3, 4, 4, 5, 7, 6, 8, 10, 10, 11, 14, 13, 16, 19, 18, 20, 25, 23, 27, 31, 30, 34, 39, 37, 42, 48, 47, 50, 59, 56, 63, 70, 68, 74, 83, 82, 89, 97, 97, 104, 116, 113, 123, 133, 133, 142, 155]
A240026
[1, 1, 2, 3, 5, 6, 10, 12, 16, 21, 27, 32, 43, 50, 60, 75, 90, 103, 128, 146, 170, 203, 234, 264, 315, 355, 402, 467, 530, 589, 684, 764, 851, 969, 1083, 1195, 1360, 1504, 1659, 1863, 2063, 2258, 2531, 2779, 3039, 3379, 3709, 4032, 4474, 4880, 5304]
A240027
[1, 1, 2, 2, 4, 4, 5, 7, 9, 9, 13, 14, 16, 20, 23, 25, 32, 34, 38, 45, 51, 55, 65, 70, 77, 89, 99, 106, 122, 131, 143, 161, 177, 189, 211, 229, 248, 272, 298, 317, 349, 378, 406, 440, 479, 511, 554, 597, 640, 686, 744]

181015(2)

Ruby


A320509 とA320510

n が50以下のとき、出力してみた。

@ary1 = [1]
@ary2 = [1]

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end

# n > 0
def f(n)
  cnt1, cnt2 = 0, 0
  partition(n, 1, n).each{|ary|
    ary << 0
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    if ary0.sort == ary0
      cnt1 += 1
      cnt2 += 1 if ary0.uniq == ary0
    end
  }
  @ary1 << cnt1
  @ary2 << cnt2
end

n = 50
(1..n).each{|i| f(i)}
puts "A320509"
p @ary1
puts "A320510"
p @ary2

出力結果
A320509
[1, 1, 2, 3, 3, 4, 6, 4, 6, 8, 7, 8, 11, 7, 12, 14, 10, 13, 19, 12, 18, 21, 16, 19, 27, 19, 25, 30, 25, 30, 37, 25, 35, 40, 35, 42, 49, 35, 49, 56, 46, 54, 66, 50, 65, 72, 60, 70, 83, 68, 84]
A320510
[1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 3, 4, 3, 4, 6, 3, 5, 6, 5, 6, 9, 5, 7, 9, 8, 8, 11, 8, 11, 13, 10, 12, 15, 11, 15, 16, 14, 16, 21, 15, 20, 22, 18, 21, 26, 21, 24, 28, 25, 28, 33]

181015

Ruby


A320387, A320385, A007294 とA179269

n が100以下のとき、出力してみた。

@ary1 = [1]
@ary2 = [1]
@ary3 = [1]
@ary4 = [1]

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
end

# n > 0
def f(n)
  cnt1, cnt2, cnt3, cnt4 = 0, 0, 0, 0
  partition(n, 1, n).each{|ary|
    ary << 0
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    ary1 = ary0.sort
    if ary1 == ary0
      cnt1 += 1
      cnt2 += 1 if ary0.uniq == ary0
    end
    if ary1 == ary0.reverse
      cnt3 += 1
      cnt4 += 1 if ary0.uniq == ary0
    end
  }
  @ary1 << cnt1
  @ary2 << cnt2
  @ary3 << cnt3
  @ary4 << cnt4
end

n = 100
(1..n).each{|i| f(i)}
puts "A320387"
p @ary1
puts "A320385"
p @ary2
puts "A007294"
p @ary3
puts "A179269"
p @ary4

出力結果
A320387
[1, 1, 1, 2, 1, 2, 3, 2, 2, 4, 3, 4, 5, 3, 5, 7, 4, 7, 8, 6, 8, 11, 7, 9, 13, 9, 11, 16, 12, 15, 18, 13, 17, 20, 17, 21, 24, 19, 24, 30, 22, 28, 34, 26, 34, 38, 30, 37, 43, 37, 42, 48, 41, 50, 58, 48, 55, 64, 53, 64, 71, 59, 73, 81, 69, 79, 89, 79, 90, 101, 87, 100, 111, 97, 115, 124, 108, 126, 137, 121, 140, 152, 138, 151, 169, 151, 166, 185, 167, 191, 201, 182, 206, 220, 204, 228, 243, 222, 250, 268, 243]
A320385
[1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 4, 3, 3, 5, 3, 5, 6, 4, 5, 7, 5, 7, 8, 6, 7, 10, 8, 9, 11, 8, 11, 13, 9, 13, 15, 12, 14, 17, 13, 16, 20, 15, 18, 22, 18, 21, 25, 20, 23, 27, 23, 28, 30, 26, 30, 34, 30, 33, 38, 31, 38, 43, 36, 42, 46, 42, 47, 50, 45, 50, 58, 51, 55, 61, 56, 64, 68, 60, 68, 74, 70, 75, 80, 74, 82, 89, 82, 89, 96, 90, 99, 104, 97, 108, 114, 107, 116]
A007294
[1, 1, 1, 2, 2, 2, 4, 4, 4, 6, 7, 7, 10, 11, 11, 15, 17, 17, 22, 24, 25, 32, 35, 36, 44, 48, 50, 60, 66, 68, 81, 89, 92, 107, 117, 121, 141, 153, 159, 181, 197, 205, 233, 252, 262, 295, 320, 332, 372, 401, 417, 465, 501, 520, 575, 619, 645, 710, 763, 793, 869, 933, 971, 1059, 1134, 1180, 1286, 1374, 1430, 1549, 1654, 1722, 1863, 1985, 2065, 2225, 2372, 2467, 2654, 2821, 2934, 3149, 3345, 3478, 3724, 3949, 4109, 4391, 4652, 4835, 5157, 5460, 5676, 6043, 6390, 6638, 7059, 7457, 7747, 8220, 8676]
A179269
[1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 7, 10, 10, 10, 13, 14, 14, 18, 19, 19, 23, 25, 25, 30, 32, 33, 38, 41, 42, 48, 52, 54, 60, 65, 67, 75, 81, 84, 92, 99, 103, 113, 121, 126, 136, 147, 153, 165, 177, 184, 197, 213, 221, 236, 253, 264, 280, 301, 313, 331, 355, 371, 390, 418, 435, 458, 490, 511, 535, 571, 595, 624, 665, 693, 723, 771, 803, 838, 891, 928, 966, 1028, 1070, 1113, 1180, 1230, 1277, 1354, 1409, 1462, 1547, 1612, 1670, 1766, 1837, 1904]

181013

Ruby


Expansion of Product_{d | ｋ} theta_3(q^d)

k <= 96 のとき、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

def B(k, n)
  ary = []
  (1..k).select{|i| k % i == 0}.each{|i|
    ary << [2 * i, 5]
    ary << [i, -2]
    ary << [4 * i, -2]
  }
  A(ary, n)
end

n = 100
(1..96).each{|i| p [i, B(i, n)]}

出力結果
[1, [1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2]]
[2, [1, 2, 2, 4, 2, 0, 4, 0, 2, 6, 0, 4, 4, 0, 0, 0, 2, 4, 6, 4, 0, 0, 4, 0, 4, 2, 0, 8, 0, 0, 0, 0, 2, 8, 4, 0, 6, 0, 4, 0, 0, 4, 0, 4, 4, 0, 0, 0, 4, 2, 2, 8, 0, 0, 8, 0, 0, 8, 0, 4, 0, 0, 0, 0, 2, 0, 8, 4, 4, 0, 0, 0, 6, 4, 0, 4, 4, 0, 0, 0, 0, 10, 4, 4, 0, 0, 4, 0, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, 2, 12, 2]]
[3, [1, 2, 0, 2, 6, 0, 0, 4, 0, 2, 0, 0, 6, 4, 0, 0, 6, 0, 0, 4, 0, 4, 0, 0, 0, 2, 0, 2, 12, 0, 0, 4, 0, 0, 0, 0, 6, 4, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 6, 6, 0, 0, 12, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 4, 6, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 2, 12, 0, 0, 4, 0, 2, 0, 0, 12, 0, 0, 0, 0, 0, 0, 8, 0, 4, 0, 0, 0, 4, 0, 0, 6]]
[4, [1, 2, 2, 4, 4, 4, 8, 8, 6, 6, 8, 4, 8, 12, 0, 8, 12, 8, 10, 12, 8, 8, 24, 8, 8, 14, 8, 16, 16, 4, 0, 16, 6, 16, 16, 8, 12, 20, 24, 8, 24, 8, 16, 20, 8, 20, 0, 16, 24, 18, 10, 8, 24, 12, 32, 24, 0, 16, 24, 12, 16, 20, 0, 24, 12, 8, 16, 28, 16, 16, 48, 8, 30, 32, 8, 20, 24, 16, 0, 16, 24, 18, 32, 20, 16, 24, 24, 24, 24, 16, 24, 24, 16, 24, 0, 8, 8, 40, 18, 20, 28]]
[5, [1, 2, 0, 0, 2, 2, 4, 0, 0, 6, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 8, 0, 0, 4, 2, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 4, 0, 0, 0, 6, 4, 0, 0, 6, 0, 0, 0, 0, 8, 0, 4, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 10, 0, 0, 8, 0, 4, 0, 0, 4, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 2]]
[6, [1, 2, 2, 6, 6, 4, 14, 8, 6, 26, 12, 16, 42, 12, 16, 44, 6, 20, 50, 16, 36, 56, 24, 16, 42, 30, 28, 78, 48, 36, 84, 40, 6, 80, 36, 48, 150, 44, 40, 100, 36, 36, 112, 48, 72, 148, 48, 48, 42, 50, 62, 124, 84, 52, 158, 64, 48, 144, 60, 64, 252, 60, 64, 200, 6, 88, 168, 64, 108, 176, 96, 80, 150, 84, 76, 218, 120, 96, 196, 72, 36, 242, 84, 80, 336, 104, 88, 204, 72, 84, 300, 112, 144, 216, 96, 128, 42, 100, 114, 304, 186]]
[7, [1, 2, 0, 0, 2, 0, 0, 2, 4, 2, 0, 4, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 2, 4, 0, 0, 8, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 10, 0, 0, 4, 0, 0, 0, 4, 4, 0, 0, 0, 0, 4, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 2]]
[8, [1, 2, 2, 4, 4, 4, 8, 8, 8, 10, 12, 12, 16, 20, 16, 24, 24, 20, 26, 20, 24, 32, 24, 24, 32, 30, 28, 40, 32, 20, 48, 32, 24, 48, 36, 48, 52, 36, 40, 56, 48, 52, 64, 44, 48, 84, 48, 48, 96, 50, 62, 72, 56, 68, 80, 72, 64, 80, 60, 60, 96, 52, 64, 104, 24, 72, 96, 68, 72, 96, 96, 72, 104, 68, 76, 124, 80, 96, 112, 80, 144, 130, 84, 84, 128, 104, 88, 120, 96, 100, 156, 112, 96, 128, 96, 120, 96, 116, 114, 156, 124]]
[9, [1, 2, 0, 2, 6, 0, 0, 4, 0, 4, 4, 0, 10, 16, 0, 0, 14, 0, 4, 4, 0, 16, 8, 0, 0, 14, 0, 2, 20, 0, 8, 4, 0, 0, 4, 0, 12, 32, 0, 8, 20, 0, 0, 12, 0, 16, 8, 0, 26, 14, 0, 0, 32, 0, 0, 8, 0, 24, 12, 0, 0, 32, 0, 8, 30, 0, 8, 12, 0, 0, 8, 0, 20, 24, 0, 10, 20, 0, 0, 12, 0, 4, 12, 0, 32, 48, 0, 0, 40, 0, 8, 8, 0, 48, 8, 0, 0, 24, 0, 8, 42]]
[10, [1, 2, 2, 4, 2, 2, 8, 4, 10, 10, 2, 16, 8, 12, 16, 4, 26, 20, 26, 24, 2, 32, 24, 28, 40, 2, 28, 40, 16, 28, 8, 24, 58, 48, 36, 4, 26, 44, 40, 56, 10, 36, 64, 36, 24, 10, 48, 52, 104, 66, 2, 72, 28, 60, 80, 16, 80, 80, 60, 56, 8, 60, 64, 116, 122, 12, 96, 76, 36, 96, 16, 72, 130, 68, 76, 4, 40, 80, 112, 80, 26, 130, 84, 68, 64, 20, 88, 120, 120, 84, 26, 120, 48, 128, 96, 24, 232, 84, 114, 144, 2]]
[11, [1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 4, 0, 0, 4, 2, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 4, 4, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 2, 6]]
[12, [1, 2, 2, 6, 8, 8, 18, 20, 18, 34, 40, 32, 56, 68, 44, 88, 108, 64, 122, 156, 96, 172, 232, 128, 234, 322, 180, 294, 400, 216, 344, 516, 234, 400, 608, 288, 488, 708, 364, 524, 864, 432, 620, 988, 480, 728, 1072, 576, 756, 1174, 594, 896, 1360, 680, 1050, 1424, 900, 1188, 1656, 896, 1248, 1892, 940, 1540, 1836, 1040, 1688, 2124, 1152, 1696, 2448, 1216, 2178, 2660, 1364, 2182, 2896, 1488, 2260, 3268, 1296, 2578, 3312, 1696, 2800, 3520, 1932, 2760, 4320, 1968, 3016, 4264, 2112, 3516, 4320, 2176, 3042, 4660, 2438, 3632, 4808]]
[13, [1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 4, 0, 2, 4, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 2, 4, 0, 0, 4, 0, 0, 0, 0, 4, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2]]
[14, [1, 2, 2, 4, 2, 0, 4, 2, 6, 10, 8, 8, 4, 8, 2, 8, 18, 12, 18, 12, 8, 4, 16, 24, 28, 30, 16, 24, 2, 20, 32, 24, 38, 32, 28, 0, 18, 20, 28, 40, 40, 28, 4, 40, 16, 48, 32, 24, 68, 2, 38, 40, 16, 28, 56, 48, 6, 48, 36, 44, 32, 48, 40, 10, 82, 56, 64, 32, 28, 80, 8, 48, 86, 36, 52, 108, 28, 8, 80, 48, 104, 66, 60, 60, 4, 80, 64, 88, 80, 68, 104, 8, 32, 96, 56, 88, 156, 60, 2, 104, 38]]
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[81, [1, 2, 0, 2, 6, 0, 0, 4, 0, 4, 4, 0, 10, 16, 0, 0, 14, 0, 4, 4, 0, 16, 8, 0, 0, 14, 0, 4, 24, 0, 12, 16, 0, 0, 12, 0, 20, 40, 0, 28, 52, 0, 0, 40, 0, 24, 16, 0, 58, 30, 0, 0, 60, 0, 4, 48, 0, 40, 20, 0, 0, 40, 0, 32, 94, 0, 24, 52, 0, 0, 32, 0, 52, 40, 0, 62, 48, 0, 0, 76, 0, 6, 32, 0, 84, 84, 0, 0, 112, 0, 32, 76, 0, 84, 64, 0, 0, 68, 0, 56, 98]]
[82, [1, 2, 2, 4, 2, 0, 4, 0, 2, 6, 0, 4, 4, 0, 0, 0, 2, 4, 6, 4, 0, 0, 4, 0, 4, 2, 0, 8, 0, 0, 0, 0, 2, 8, 4, 0, 6, 0, 4, 0, 0, 6, 4, 8, 12, 4, 0, 8, 4, 6, 14, 8, 8, 8, 8, 0, 0, 12, 8, 16, 8, 0, 0, 8, 2, 8, 12, 4, 20, 0, 0, 0, 6, 8, 16, 12, 4, 12, 0, 8, 0, 10, 14, 8, 12, 16, 8, 0, 12, 12, 8, 16, 16, 8, 8, 16, 4, 4, 22, 20, 22]]
[83, [1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 4, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 2]]
[84, [1, 2, 2, 6, 8, 8, 18, 22, 22, 38, 52, 48, 72, 104, 86, 128, 180, 156, 202, 284, 268, 306, 456, 424, 478, 678, 652, 730, 960, 980, 1036, 1400, 1378, 1472, 1940, 1864, 2072, 2588, 2616, 2688, 3472, 3500, 3558, 4672, 4524, 4832, 5896, 6024, 6236, 7498, 7766, 7876, 9864, 9564, 10186, 12152, 12002, 12844, 15092, 15316, 15324, 19040, 18520, 19282, 23288, 22584, 24024, 27560, 28244, 28328, 33788, 33440, 34382, 41444, 39212, 41922, 47800, 47824, 49204, 57672, 57124, 56650, 68932, 65748, 68928, 79272, 78344, 80652, 91660, 92852, 91500, 108840, 104980, 109336, 126256, 121320, 126490, 141612, 143402, 142816, 167800]]
[85, [1, 2, 0, 0, 2, 2, 4, 0, 0, 6, 0, 0, 0, 0, 4, 0, 2, 2, 4, 0, 2, 12, 4, 8, 4, 2, 12, 0, 0, 4, 4, 8, 0, 4, 0, 0, 6, 4, 16, 0, 0, 12, 4, 0, 0, 6, 12, 8, 0, 6, 0, 0, 0, 12, 8, 0, 4, 0, 8, 0, 0, 4, 12, 8, 2, 0, 12, 0, 2, 12, 4, 16, 4, 12, 8, 0, 0, 12, 8, 0, 2, 14, 8, 0, 12, 2, 24, 8, 4, 24, 4, 8, 8, 4, 16, 0, 4, 12, 28, 8, 2]]
[86, [1, 2, 2, 4, 2, 0, 4, 0, 2, 6, 0, 4, 4, 0, 0, 0, 2, 4, 6, 4, 0, 0, 4, 0, 4, 2, 0, 8, 0, 0, 0, 0, 2, 8, 4, 0, 6, 0, 4, 0, 0, 4, 0, 6, 8, 4, 8, 4, 4, 10, 2, 12, 12, 0, 16, 8, 0, 8, 0, 8, 8, 12, 8, 0, 2, 8, 8, 12, 8, 0, 16, 0, 6, 4, 0, 8, 20, 8, 0, 12, 0, 18, 4, 4, 8, 0, 14, 12, 8, 12, 4, 8, 12, 4, 20, 12, 4, 28, 10, 12, 18]]
[87, [1, 2, 0, 2, 6, 0, 0, 4, 0, 2, 0, 0, 6, 4, 0, 0, 6, 0, 0, 4, 0, 4, 0, 0, 0, 2, 0, 2, 12, 2, 4, 4, 4, 12, 0, 0, 14, 4, 4, 4, 0, 12, 8, 4, 0, 12, 0, 0, 14, 6, 8, 0, 12, 0, 4, 0, 4, 28, 0, 0, 8, 4, 0, 4, 6, 12, 8, 4, 8, 0, 0, 0, 8, 4, 0, 2, 12, 12, 12, 4, 0, 26, 0, 0, 12, 0, 8, 2, 4, 0, 12, 20, 8, 16, 8, 0, 12, 4, 0, 12, 14]]
[88, [1, 2, 2, 4, 4, 4, 8, 8, 8, 10, 12, 14, 20, 24, 24, 32, 32, 36, 42, 36, 44, 56, 50, 60, 76, 70, 84, 96, 88, 88, 104, 100, 112, 124, 124, 160, 160, 156, 184, 192, 172, 220, 216, 204, 244, 276, 296, 292, 336, 322, 350, 344, 368, 396, 392, 412, 472, 488, 460, 548, 548, 528, 520, 624, 552, 592, 680, 684, 744, 736, 792, 812, 828, 780, 940, 868, 880, 1016, 1096, 1072, 1136, 1306, 1164, 1220, 1224, 1312, 1312, 1368, 1448, 1532, 1596, 1632, 1764, 1728, 1632, 1680, 1912, 1692, 1842, 2086, 2040]]
[89, [1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 4, 0, 0, 0, 0, 4, 0, 2]]
[90, [1, 2, 2, 6, 6, 6, 18, 12, 18, 40, 26, 52, 74, 48, 92, 118, 102, 168, 200, 184, 278, 332, 292, 384, 498, 470, 640, 798, 636, 888, 1122, 936, 1250, 1520, 1356, 1884, 2280, 1760, 2280, 2884, 2498, 3332, 3904, 3184, 4324, 5288, 4160, 5088, 6258, 5578, 7274, 8764, 6728, 8152, 10826, 8596, 10436, 13096, 10380, 13972, 17338, 12644, 15128, 19408, 16214, 20248, 24952, 19304, 23548, 30804, 23452, 26616, 34904, 28036, 35000, 44998, 33000, 38104, 50492, 39784, 46134, 60644, 46864, 55496, 74608, 54440, 61264, 81364, 62972, 76300, 99816, 74752, 84592, 112360, 87456, 96536, 128578, 100204, 113066, 157856, 115814]]
[91, [1, 2, 0, 0, 2, 0, 0, 2, 4, 2, 0, 4, 0, 2, 4, 0, 6, 4, 0, 0, 4, 8, 4, 4, 8, 2, 0, 0, 2, 16, 0, 0, 8, 0, 0, 0, 10, 4, 4, 0, 0, 4, 8, 4, 4, 16, 0, 0, 0, 6, 8, 0, 2, 8, 0, 0, 16, 8, 0, 4, 8, 4, 4, 10, 10, 0, 8, 4, 12, 8, 0, 4, 4, 0, 0, 8, 4, 28, 0, 4, 12, 10, 0, 0, 24, 8, 0, 0, 12, 8, 8, 2, 16, 0, 4, 12, 8, 0, 4, 12, 6]]
[92, [1, 2, 2, 4, 4, 4, 8, 8, 6, 6, 8, 4, 8, 12, 0, 8, 12, 8, 10, 12, 8, 8, 24, 10, 12, 18, 16, 24, 24, 20, 16, 28, 18, 32, 24, 24, 36, 20, 40, 32, 40, 28, 40, 36, 24, 68, 18, 36, 56, 42, 50, 48, 48, 28, 76, 48, 48, 56, 56, 60, 56, 84, 40, 88, 48, 64, 72, 60, 104, 36, 104, 92, 98, 100, 72, 108, 80, 136, 84, 80, 104, 114, 128, 92, 136, 88, 152, 104, 120, 120, 128, 192, 84, 192, 128, 152, 172, 152, 154, 124, 212]]
[93, [1, 2, 0, 2, 6, 0, 0, 4, 0, 2, 0, 0, 6, 4, 0, 0, 6, 0, 0, 4, 0, 4, 0, 0, 0, 2, 0, 2, 12, 0, 0, 6, 4, 0, 4, 12, 6, 4, 8, 4, 4, 0, 0, 16, 8, 0, 0, 12, 6, 6, 8, 0, 20, 0, 0, 0, 4, 4, 4, 24, 0, 4, 8, 4, 6, 0, 0, 16, 8, 0, 8, 0, 0, 4, 8, 2, 12, 0, 0, 16, 12, 2, 0, 24, 12, 0, 0, 0, 8, 0, 0, 8, 8, 6, 12, 12, 4, 16, 8, 0, 14]]
[94, [1, 2, 2, 4, 2, 0, 4, 0, 2, 6, 0, 4, 4, 0, 0, 0, 2, 4, 6, 4, 0, 0, 4, 0, 4, 2, 0, 8, 0, 0, 0, 0, 2, 8, 4, 0, 6, 0, 4, 0, 0, 4, 0, 4, 4, 0, 0, 2, 8, 6, 10, 12, 0, 8, 8, 4, 12, 8, 8, 12, 0, 0, 0, 4, 10, 12, 16, 4, 4, 8, 0, 8, 10, 4, 16, 4, 4, 0, 0, 4, 16, 18, 4, 16, 0, 8, 4, 0, 12, 4, 8, 8, 0, 0, 2, 12, 12, 16, 22, 12, 10]]
[95, [1, 2, 0, 0, 2, 2, 4, 0, 0, 6, 0, 0, 0, 0, 4, 0, 2, 0, 0, 2, 6, 8, 0, 4, 8, 10, 0, 0, 12, 4, 4, 0, 0, 8, 0, 4, 6, 0, 0, 4, 16, 4, 0, 8, 4, 6, 4, 0, 8, 14, 0, 0, 0, 0, 8, 12, 4, 0, 0, 0, 8, 4, 0, 0, 14, 8, 0, 0, 12, 8, 4, 0, 0, 16, 0, 8, 2, 4, 0, 0, 14, 14, 8, 4, 8, 12, 4, 0, 16, 12, 8, 0, 4, 0, 4, 2, 12, 16, 0, 8, 34]]
[96, [1, 2, 2, 6, 8, 8, 18, 20, 20, 38, 44, 44, 72, 84, 80, 128, 146, 136, 206, 232, 224, 324, 356, 344, 492, 530, 516, 706, 752, 744, 1004, 1068, 1034, 1392, 1468, 1428, 1880, 1956, 1924, 2500, 2600, 2548, 3264, 3392, 3344, 4216, 4376, 4288, 5382, 5590, 5462, 6824, 7056, 6912, 8518, 8808, 8624, 10540, 10908, 10688, 12992, 13444, 13128, 15820, 16398, 16000, 19244, 19848, 19344, 23200, 23904, 23344, 27700, 28572, 27908, 32998, 34000, 33056, 39060, 40236, 39172, 45982, 47244, 45956, 53840, 55296, 53788, 62816, 64400, 62752, 72868, 74712, 72640, 84260, 86304, 83888, 97238, 99216, 96458, 111472, 113752]]

181007(2)

Ruby


Ramanujan's 54 universal quaternary quadratic forms(2)

Conway–Schneeberger Fifteen Theorem を使えば、

1から15までの表し方が存在していれば、16以上も表し方が存在。

このことを利用して、コードをシンプルにしてみた。

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

def B(a, n)
  ary = []
  a.each{|i|
    ary << [2 * i, 5]
    ary << [i, -2]
    ary << [4 * i, -2]
  }
  A(ary, n)
end

n = 100
s = 0
(1..20).to_a.repeated_combination(3){|a|
  a.unshift(1)
  if B(a, 15).select{|i| i == 0} == []
    s += 1
    p [s, a]
    p B(a, n)
  end
}

出力結果
[1, [1, 1, 1, 1]]
[1, 8, 24, 32, 24, 48, 96, 64, 24, 104, 144, 96, 96, 112, 192, 192, 24, 144, 312, 160, 144, 256, 288, 192, 96, 248, 336, 320, 192, 240, 576, 256, 24, 384, 432, 384, 312, 304, 480, 448, 144, 336, 768, 352, 288, 624, 576, 384, 96, 456, 744, 576, 336, 432, 960, 576, 192, 640, 720, 480, 576, 496, 768, 832, 24, 672, 1152, 544, 432, 768, 1152, 576, 312, 592, 912, 992, 480, 768, 1344, 640, 144, 968, 1008, 672, 768, 864, 1056, 960, 288, 720, 1872, 896, 576, 1024, 1152, 960, 96, 784, 1368, 1248, 744]
[2, [1, 1, 1, 2]]
[1, 6, 14, 20, 30, 40, 36, 48, 62, 42, 72, 100, 68, 120, 112, 48, 126, 108, 98, 180, 136, 160, 180, 144, 132, 126, 216, 200, 240, 280, 112, 192, 254, 120, 252, 320, 210, 360, 324, 144, 264, 252, 288, 420, 340, 280, 336, 288, 260, 342, 294, 360, 408, 520, 360, 240, 496, 216, 504, 580, 240, 600, 448, 336, 510, 288, 280, 660, 540, 480, 576, 432, 434, 444, 648, 420, 612, 800, 336, 480, 520, 366, 588, 820, 544, 720, 756, 336, 660, 540, 504, 960, 720, 640, 672, 432, 516, 588, 798, 700, 630]
[3, [1, 1, 1, 3]]
[1, 6, 12, 10, 18, 48, 40, 12, 60, 78, 24, 48, 70, 84, 120, 32, 66, 192, 84, 36, 144, 180, 120, 96, 136, 126, 168, 82, 84, 336, 200, 60, 252, 288, 96, 96, 234, 228, 360, 140, 120, 480, 144, 84, 336, 336, 240, 192, 310, 258, 252, 128, 252, 624, 400, 96, 408, 540, 168, 240, 224, 372, 600, 156, 258, 672, 360, 132, 576, 576, 240, 288, 420, 444, 456, 210, 252, 864, 560, 156, 528, 726, 240, 336, 540, 384, 840, 224, 408, 1056, 312, 168, 672, 900, 480, 288, 520, 588, 516, 336, 378]
[4, [1, 1, 1, 4]]
[1, 6, 12, 8, 8, 36, 48, 16, 24, 78, 72, 24, 32, 84, 96, 48, 24, 108, 156, 40, 48, 192, 144, 48, 96, 186, 168, 80, 64, 180, 288, 64, 24, 288, 216, 96, 104, 228, 240, 112, 144, 252, 384, 88, 96, 468, 288, 96, 96, 342, 372, 144, 112, 324, 480, 144, 192, 480, 360, 120, 192, 372, 384, 208, 24, 504, 576, 136, 144, 576, 576, 144, 312, 444, 456, 248, 160, 576, 672, 160, 144, 726, 504, 168, 256, 648, 528, 240, 288, 540, 936, 224, 192, 768, 576, 240, 96, 588, 684, 312, 248]
[5, [1, 1, 1, 5]]
[1, 6, 12, 8, 6, 26, 36, 24, 28, 42, 72, 72, 8, 48, 108, 48, 54, 64, 84, 120, 26, 72, 144, 88, 84, 126, 216, 80, 24, 180, 156, 192, 92, 96, 288, 144, 42, 144, 240, 144, 168, 252, 144, 168, 72, 182, 396, 184, 72, 258, 372, 192, 48, 208, 360, 312, 252, 160, 360, 360, 48, 372, 384, 168, 246, 288, 432, 264, 64, 264, 468, 432, 196, 288, 648, 248, 120, 288, 288, 480, 234, 366, 504, 328, 72, 384, 756, 240, 336, 540, 504, 432, 88, 256, 828, 520, 276, 384, 516, 504, 126]
[6, [1, 1, 1, 6]]
[1, 6, 12, 8, 6, 24, 26, 12, 36, 46, 36, 72, 56, 24, 72, 60, 54, 96, 52, 72, 120, 48, 36, 96, 98, 90, 144, 144, 60, 120, 144, 60, 180, 172, 96, 240, 142, 72, 192, 96, 132, 288, 180, 120, 216, 168, 144, 192, 152, 162, 276, 256, 120, 216, 210, 120, 264, 256, 180, 360, 300, 120, 360, 156, 150, 480, 208, 264, 384, 240, 168, 288, 316, 252, 432, 440, 216, 240, 360, 156, 504, 374, 144, 552, 240, 192, 384, 300, 228, 576, 468, 240, 480, 240, 288, 480, 434, 300, 420, 440, 378]
[7, [1, 1, 1, 7]]
[1, 6, 12, 8, 6, 24, 24, 2, 24, 54, 40, 36, 56, 72, 48, 24, 66, 96, 84, 40, 72, 144, 24, 12, 120, 102, 120, 80, 98, 132, 72, 64, 84, 240, 160, 48, 198, 180, 120, 72, 136, 240, 240, 84, 84, 312, 120, 96, 248, 246, 180, 96, 216, 228, 240, 80, 156, 360, 216, 120, 168, 360, 192, 194, 174, 216, 400, 36, 288, 528, 144, 84, 336, 432, 312, 168, 280, 396, 216, 180, 264, 558, 400, 168, 432, 288, 168, 240, 288, 528, 520, 144, 252, 576, 288, 120, 504, 576, 396, 132, 246]
[8, [1, 1, 2, 2]]
[1, 4, 8, 16, 24, 24, 32, 32, 24, 52, 48, 48, 96, 56, 64, 96, 24, 72, 104, 80, 144, 128, 96, 96, 96, 124, 112, 160, 192, 120, 192, 128, 24, 192, 144, 192, 312, 152, 160, 224, 144, 168, 256, 176, 288, 312, 192, 192, 96, 228, 248, 288, 336, 216, 320, 288, 192, 320, 240, 240, 576, 248, 256, 416, 24, 336, 384, 272, 432, 384, 384, 288, 312, 296, 304, 496, 480, 384, 448, 320, 144, 484, 336, 336, 768, 432, 352, 480, 288, 360, 624, 448, 576, 512, 384, 480, 96, 392, 456, 624, 744]
[9, [1, 1, 2, 3]]
[1, 4, 6, 10, 20, 20, 24, 40, 22, 28, 56, 20, 50, 80, 28, 80, 84, 32, 78, 80, 68, 100, 120, 80, 88, 124, 56, 82, 136, 100, 140, 200, 86, 80, 192, 72, 140, 240, 120, 200, 248, 80, 112, 176, 100, 260, 224, 160, 210, 172, 186, 128, 272, 180, 240, 400, 124, 200, 280, 116, 272, 400, 140, 280, 340, 144, 300, 272, 160, 320, 432, 240, 286, 296, 168, 310, 400, 200, 224, 520, 260, 244, 480, 164, 340, 640, 264, 400, 440, 176, 392, 288, 272, 500, 448, 400, 344, 392, 258, 260, 620]
[10, [1, 1, 2, 4]]
[1, 4, 6, 8, 14, 16, 20, 32, 30, 28, 40, 40, 36, 48, 48, 32, 62, 72, 42, 72, 72, 64, 100, 96, 68, 84, 120, 80, 112, 112, 48, 128, 126, 80, 108, 128, 98, 144, 180, 96, 136, 168, 160, 168, 180, 112, 144, 192, 132, 228, 126, 144, 216, 208, 200, 160, 240, 144, 280, 232, 112, 240, 192, 224, 254, 192, 120, 264, 252, 192, 320, 288, 210, 296, 360, 168, 324, 320, 144, 320, 264, 244, 252, 328, 288, 288, 420, 224, 340, 360, 280, 384, 336, 256, 288, 288, 260, 392, 342, 280, 294]
[11, [1, 1, 2, 5]]
[1, 4, 6, 8, 12, 10, 16, 28, 22, 36, 40, 24, 56, 36, 24, 64, 28, 64, 78, 24, 58, 48, 68, 92, 80, 92, 72, 112, 92, 48, 104, 96, 118, 176, 64, 48, 124, 84, 148, 160, 104, 120, 176, 120, 72, 146, 88, 204, 216, 124, 126, 96, 180, 148, 224, 188, 120, 304, 216, 120, 224, 96, 192, 332, 156, 240, 160, 168, 224, 144, 196, 192, 350, 304, 168, 232, 168, 168, 336, 224, 234, 340, 252, 200, 192, 192, 240, 480, 292, 264, 392, 144, 316, 320, 184, 332, 304, 416, 258, 216, 308]
[12, [1, 1, 2, 6]]
[1, 4, 6, 8, 12, 8, 10, 24, 18, 28, 48, 24, 40, 56, 12, 40, 60, 32, 78, 72, 24, 48, 48, 48, 70, 84, 84, 80, 120, 56, 32, 120, 66, 120, 192, 48, 84, 152, 36, 112, 144, 80, 180, 168, 120, 104, 96, 96, 136, 172, 126, 160, 168, 104, 82, 192, 84, 144, 336, 120, 200, 248, 60, 168, 252, 112, 288, 264, 96, 240, 96, 144, 234, 296, 228, 168, 360, 144, 140, 312, 120, 244, 480, 168, 144, 256, 84, 280, 336, 176, 336, 336, 240, 240, 192, 144, 310, 392, 258, 312, 252]
[13, [1, 1, 2, 7]]
[1, 4, 6, 8, 12, 8, 8, 18, 14, 24, 40, 32, 40, 40, 32, 28, 36, 64, 46, 72, 72, 16, 56, 40, 40, 88, 72, 80, 82, 64, 44, 64, 86, 96, 128, 144, 64, 64, 136, 44, 136, 160, 80, 144, 128, 120, 92, 128, 104, 164, 158, 96, 168, 80, 144, 160, 168, 108, 104, 232, 108, 168, 192, 82, 156, 252, 160, 208, 192, 80, 272, 132, 134, 288, 152, 248, 232, 200, 124, 148, 328, 192, 288, 328, 144, 160, 264, 160, 248, 352, 216, 336, 184, 128, 256, 188, 232, 384, 278, 240, 320]
[14, [1, 1, 2, 8]]
[1, 4, 6, 8, 12, 8, 8, 16, 8, 20, 36, 24, 48, 40, 16, 48, 24, 40, 78, 40, 72, 64, 24, 48, 32, 60, 84, 80, 96, 40, 48, 64, 24, 96, 108, 96, 156, 72, 40, 112, 48, 104, 192, 88, 144, 168, 48, 96, 96, 100, 186, 144, 168, 136, 80, 144, 64, 160, 180, 120, 288, 104, 64, 208, 24, 144, 288, 136, 216, 192, 96, 144, 104, 136, 228, 248, 240, 192, 112, 160, 144, 260, 252, 168, 384, 208, 88, 240, 96, 200, 468, 224, 288, 256, 96, 240, 96, 232, 342, 312, 372]
[15, [1, 1, 2, 9]]
[1, 4, 6, 8, 12, 8, 8, 16, 6, 14, 32, 20, 40, 48, 16, 32, 44, 28, 54, 72, 40, 64, 72, 16, 40, 52, 56, 92, 96, 56, 32, 80, 38, 48, 104, 64, 102, 144, 52, 32, 112, 44, 128, 168, 68, 136, 128, 96, 104, 132, 62, 144, 176, 104, 172, 96, 112, 112, 192, 116, 96, 240, 96, 176, 204, 32, 112, 264, 108, 192, 224, 48, 198, 168, 104, 168, 232, 160, 160, 208, 104, 178, 232, 164, 192, 288, 164, 96, 264, 140, 232, 384, 144, 256, 288, 112, 232, 232, 150, 340, 252]
[16, [1, 1, 2, 10]]
[1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 26, 16, 36, 40, 24, 32, 28, 48, 42, 48, 72, 32, 72, 64, 8, 60, 48, 64, 108, 56, 48, 64, 54, 64, 64, 72, 84, 104, 120, 32, 26, 88, 72, 120, 144, 88, 88, 144, 84, 92, 126, 64, 216, 152, 80, 160, 24, 128, 180, 144, 156, 104, 192, 128, 92, 168, 96, 184, 288, 96, 144, 160, 42, 224, 144, 104, 240, 224, 144, 128, 168, 100, 252, 248, 144, 208, 168, 176, 72, 168, 182, 160, 396, 192, 184, 224, 72, 272, 258, 176, 372]
[17, [1, 1, 2, 11]]
[1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 24, 10, 32, 36, 16, 40, 28, 32, 62, 36, 48, 64, 40, 64, 56, 28, 56, 56, 80, 68, 48, 80, 38, 80, 80, 32, 92, 88, 88, 112, 40, 16, 112, 52, 90, 144, 44, 120, 128, 100, 78, 96, 116, 144, 160, 112, 128, 80, 136, 104, 144, 132, 92, 240, 108, 112, 192, 88, 128, 232, 144, 176, 206, 80, 120, 128, 116, 192, 208, 160, 160, 212, 176, 132, 224, 128, 200, 272, 136, 112, 136, 160, 168, 312, 140, 144, 248, 176, 214, 186, 156]
[18, [1, 1, 2, 12]]
[1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 24, 8, 26, 32, 12, 32, 36, 32, 46, 56, 36, 40, 72, 32, 56, 76, 24, 64, 72, 40, 60, 80, 54, 64, 96, 48, 52, 96, 72, 80, 120, 32, 48, 104, 36, 104, 96, 64, 98, 124, 90, 64, 144, 72, 144, 160, 60, 136, 120, 104, 144, 160, 60, 112, 180, 96, 172, 200, 96, 128, 240, 96, 142, 200, 72, 200, 192, 80, 96, 208, 132, 164, 288, 104, 180, 256, 120, 160, 216, 128, 168, 192, 144, 200, 192, 160, 152, 248, 162, 168, 276]
[19, [1, 1, 2, 13]]
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[20, [1, 1, 2, 14]]
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[21, [1, 1, 3, 3]]
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[22, [1, 1, 3, 4]]
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[23, [1, 1, 3, 5]]
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[24, [1, 1, 3, 6]]
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[25, [1, 2, 2, 2]]
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[26, [1, 2, 2, 3]]
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[27, [1, 2, 2, 4]]
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[28, [1, 2, 2, 5]]
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[29, [1, 2, 2, 6]]
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[30, [1, 2, 2, 7]]
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[31, [1, 2, 3, 3]]
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[32, [1, 2, 3, 4]]
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[33, [1, 2, 3, 5]]
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[34, [1, 2, 3, 6]]
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[35, [1, 2, 3, 7]]
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[36, [1, 2, 3, 8]]
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[37, [1, 2, 3, 9]]
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[38, [1, 2, 3, 10]]
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[39, [1, 2, 4, 4]]
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[40, [1, 2, 4, 5]]
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[41, [1, 2, 4, 6]]
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[42, [1, 2, 4, 7]]
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[43, [1, 2, 4, 8]]
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[44, [1, 2, 4, 9]]
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[45, [1, 2, 4, 10]]
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[46, [1, 2, 4, 11]]
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[47, [1, 2, 4, 12]]
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[48, [1, 2, 4, 13]]
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[49, [1, 2, 4, 14]]
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[50, [1, 2, 5, 6]]
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[51, [1, 2, 5, 7]]
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[52, [1, 2, 5, 8]]
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[53, [1, 2, 5, 9]]
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[54, [1, 2, 5, 10]]
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