{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "kが無限のとき、%o1の式を無限級数で表すと、次数の数を２つの３乗数の和で表す方法の個数がその次数の項の係数になります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{3}$}\\left(\\sum_{n=1}^{k}{x^{n^3}}\\right)^2\\]"
      ],
      "text/plain": [
       "                                   k\n",
       "                                  ====    3\n",
       "                                  \\      n  2\n",
       "(%o3)                            ( >    x  )\n",
       "                                  /\n",
       "                                  ====\n",
       "                                  n = 1"
      ],
      "text/x-maxima": [
       "('sum(x^n^3,n,1,k))^2"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SN:sum(x^(n^3),n,1,k)^2;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "kを有限にしても$(k^3+1)$次の項までの係数は正しく計算されます。k=12としてやってみましょう。下記の式では係数の不正確な$x^{k^3+1}$より次数の大きな項は捨てています。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{4}$}4\\,x^{1729}+2\\,x^{1674}+2\\,x^{1547}+2\\,x^{1512}+x^{1458}+2\\,x^{1456}+2\\,x^{1395}+2\\,x^{1358}+2\\,x^{1343}+2\\,x^{1339}+2\\,x^{1332}+2\\,x^{1241}+2\\,x^{1216}+2\\,x^{1125}+2\\,x^{1072}+2\\,x^{1064}+2\\,x^{1027}+x^{1024}+2\\,x^{1008}+2\\,x^{1001}+2\\,x^{945}+2\\,x^{855}+2\\,x^{854}+2\\,x^{793}+2\\,x^{756}+2\\,x^{737}+2\\,x^{730}+2\\,x^{728}+x^{686}+2\\,x^{637}+2\\,x^{576}+2\\,x^{559}+2\\,x^{539}+2\\,x^{520}+2\\,x^{513}+2\\,x^{468}+x^{432}+2\\,x^{407}+2\\,x^{370}+2\\,x^{351}+2\\,x^{344}+2\\,x^{341}+2\\,x^{280}+x^{250}+2\\,x^{243}+2\\,x^{224}+2\\,x^{217}+2\\,x^{189}+2\\,x^{152}+2\\,x^{133}+x^{128}+2\\,x^{126}+2\\,x^{91}+2\\,x^{72}+2\\,x^{65}+x^{54}+2\\,x^{35}+2\\,x^{28}+x^{16}+2\\,x^9+x^2\\]"
      ],
      "text/plain": [
       "         1729      1674      1547      1512    1458      1456      1395\n",
       "(%o4) 4 x     + 2 x     + 2 x     + 2 x     + x     + 2 x     + 2 x\n",
       "      1358      1343      1339      1332      1241      1216      1125\n",
       " + 2 x     + 2 x     + 2 x     + 2 x     + 2 x     + 2 x     + 2 x\n",
       "      1072      1064      1027    1024      1008      1001      945      855\n",
       " + 2 x     + 2 x     + 2 x     + x     + 2 x     + 2 x     + 2 x    + 2 x\n",
       "      854      793      756      737      730      728    686      637      576\n",
       " + 2 x    + 2 x    + 2 x    + 2 x    + 2 x    + 2 x    + x    + 2 x    + 2 x\n",
       "      559      539      520      513      468    432      407      370      351\n",
       " + 2 x    + 2 x    + 2 x    + 2 x    + 2 x    + x    + 2 x    + 2 x    + 2 x\n",
       "      344      341      280    250      243      224      217      189      152\n",
       " + 2 x    + 2 x    + 2 x    + x    + 2 x    + 2 x    + 2 x    + 2 x    + 2 x\n",
       "      133    128      126      91      72      65    54      35      28    16\n",
       " + 2 x    + x    + 2 x    + 2 x   + 2 x   + 2 x   + x   + 2 x   + 2 x   + x\n",
       "      9    2\n",
       " + 2 x  + x"
      ],
      "text/x-maxima": [
       "4*x^1729+2*x^1674+2*x^1547+2*x^1512+x^1458+2*x^1456+2*x^1395+2*x^1358+2*x^1343\n",
       "        +2*x^1339+2*x^1332+2*x^1241+2*x^1216+2*x^1125+2*x^1072+2*x^1064\n",
       "        +2*x^1027+x^1024+2*x^1008+2*x^1001+2*x^945+2*x^855+2*x^854+2*x^793\n",
       "        +2*x^756+2*x^737+2*x^730+2*x^728+x^686+2*x^637+2*x^576+2*x^559+2*x^539\n",
       "        +2*x^520+2*x^513+2*x^468+x^432+2*x^407+2*x^370+2*x^351+2*x^344+2*x^341\n",
       "        +2*x^280+x^250+2*x^243+2*x^224+2*x^217+2*x^189+2*x^152+2*x^133+x^128\n",
       "        +2*x^126+2*x^91+2*x^72+2*x^65+x^54+2*x^35+2*x^28+x^16+2*x^9+x^2"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TAXI:remainder(ev(SN,nouns),x^(k^3+2)),k:12;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$x^{1729}$の係数が4になっていることは、$1729=12^3+1^3=1^3+12^3=10^3+9^3=9^3+10^3$と４通りに表すことができることに対応しています。\n",
    "$x^{16}$の係数が１であることは、$16=2^3+2^3$と１通りに表すことができることに対応しています。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "下記のコードで、上記の多項式で係数が4以上の項の次数を取り出すことが出来ます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{5}$}\\left[ 1729 \\right] \\]"
      ],
      "text/plain": [
       "(%o5)                               [1729]"
      ],
      "text/x-maxima": [
       "[1729]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "apply(append,map(lambda([term],\n",
    "                    block([cf:coeff(term,x^hipow(term,x))],\n",
    "                          if cf>3 then [hipow(term,x)] \n",
    "                          else [])),\n",
    "                 args(TAXI)));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "kの値を大きくすると%o1をそのまま計算するコードはどんどん遅くなります。それを避けるために、与えられたxの多項式polyの２乗を指定された次数maxdまで計算する関数\n",
    "\n",
    "sqexpand(poly, maxd)\n",
    "\n",
    "を下記のように定義します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{6}$}{\\it sqexpand}\\left({\\it poly} , {\\it maxd}\\right):=\\mathbf{block}\\;\\left(\\left[ {\\it tlist}:{\\it args}\\left({\\it poly}\\right) , {\\it res}:0 \\right]  , \\mathbf{for}\\;{\\it term}\\;\\mathbf{in}\\;{\\it tlist}\\;\\mathbf{do}\\;{\\it res}:{\\it remainder}\\left({\\it res}+{\\it expand}\\left({\\it term}\\,{\\it poly}\\right) , x^{{\\it maxd}+1}\\right) , {\\it return}\\left({\\it res}\\right)\\right)\\]"
      ],
      "text/plain": [
       "(%o6) sqexpand(poly, maxd) := block([tlist : args(poly), res : 0], \n",
       "                                                               maxd + 1\n",
       "for term in tlist do res : remainder(res + expand(term poly), x        ), \n",
       "return(res))"
      ],
      "text/x-maxima": [
       "sqexpand(poly,maxd):=block([tlist:args(poly),res:0],\n",
       "         for term in tlist do res:remainder(res+expand(term*poly),x^(maxd+1)),\n",
       "         return(res))"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqexpand(poly,maxd):=block([tlist:args(poly),res:0],\n",
    "  for term in tlist do res:remainder(res+expand(term*poly),x^(maxd+1)),\n",
    "  return(res));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "早速この関数を使ってk=450まで、次数としては$450^3+1$まで求めてみます。ただし出力項数が多すぎるため表示しません。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "TT:sqexpand(sum(x^(n^3),n,1,450),450^3+1)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "結果を見るために、下記のコードを使い、係数が4以上の項の次数を出力します。また係数が５以上の項については係数と次数を出力します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{8}$}\\left[ 91034307 , 90527229 , 90091008 , 89576767 , 88780536 , 88129323 , 88122125 , 87952501 , 87699456 , 87579037 , \\left[ \\mbox{ ** } , 6 , \\mbox{ ** } , 87539319 \\right]  , 87483968 , 87029432 , 86960952 , 86575168 , 86488136 , 86368464 , 86316741 , 86124824 , 85622264 , 85502375 , 85492792 , 85188096 , 84717568 , 84637287 , 84266000 , 83758528 , 83393856 , 81621568 , 80779032 , 80668224 , 80294760 , 80208576 , 80102574 , 79692193 , 79598125 , 77414967 , 77335776 , 77129864 , 76487168 , 75550088 , 74130875 , 74097261 , 74093832 , 73842713 , 73819863 , 73369088 , 72669177 , 72131904 , 71509312 , 71292312 , 70979896 , 70957971 , 70796808 , 70449093 , 69934592 , 69805125 , 69625969 , 69583995 , 69177024 , 68802048 , 68696000 , 68007673 , 67956616 , 67944312 , 67931136 , 67173463 , 66763333 , 66469429 , 66201408 , 65728000 , 65573928 , 65293317 , 65272753 , 65055744 , 64623104 , 64421875 , 64232000 , 64125000 , 63660032 , 63057960 , 62980288 , 62135073 , 62108397 , 61568208 , 61555832 , 61121024 , 60965288 , 60698521 , 60271939 , 59913728 , 58923072 , 58751784 , 58438072 , 57066633 , 56754152 , 56733696 , 56655872 , 56388339 , 55486125 , 55306557 , 54887625 , 54814509 , 54269091 , 54267003 , 53936064 , 53283923 , 53226297 , 52739288 , 52324272 , 52242463 , 52136000 , 51564032 , 51508639 , 51479064 , 51278929 , 51240427 , 50324625 , 49933368 , 49849856 , 49844032 , 49832523 , 48469976 , 48316662 , 48137544 , 48104000 , 47915712 , 46825128 , 46683000 , 46661832 , 46394712 , 46343059 , 46179224 , 46091997 , 45882739 , 45440551 , 45118528 , 44409995 , 43997625 , 43946000 , 43852536 , 43777216 , 43699392 , 43570872 , 43450344 , 42549416 , 42301656 , 42276843 , 42168581 , 42154840 , 41892515 , 41511232 , 41436873 , 41400639 , 41343640 , 41301953 , 41258737 , 41080832 , 40970375 , 40519115 , 40354119 , 40183262 , 40134584 , 40033000 , 39639691 , 39448808 , 39312000 , 38253878 , 38007144 , 37955008 , 37498032 , 37384704 , 36998208 , 36856512 , 36094464 , 36050904 , 36047592 , 35938728 , 35740224 , 34743312 , 34031907 , 33652736 , 33394248 , 33108264 , 32984000 , 32973759 , 32886784 , 32832000 , 32530680 , 32268024 , 32114143 , 31256000 , 30511936 , 30388904 , 29791125 , 29503656 , 29184057 , 29025864 , 28981125 , 28703269 , 28699272 , 28658448 , 28408896 , 28149336 , 28102464 , 28094976 , 27882127 , 27831951 , 27529073 , 27445392 , 27262872 , 27015625 , 27003375 , 27000064 , 27000008 , 26856576 , 26569809 , 26506467 , 26122131 , 25937576 , 25785432 , 25766208 , 25371171 , 25164405 , 24897817 , 24858171 , 24767171 , 24666560 , 24480125 , 23934528 , 23901696 , 22914304 , 22754277 , 22544704 , 22543417 , 22426369 , 22031576 , 21994875 , 21753576 , 21531608 , 21411000 , 21125384 , 21036743 , 20976832 , 20683000 , 20616463 , 20496896 , 20162952 , 20127744 , 20072017 , 19346168 , 19250163 , 19034379 , 18673625 , 18617625 , 18468513 , 18410392 , 18406603 , 17999072 , 17045567 , 16983854 , 16809984 , 16797375 , 16776487 , 16587207 , 16387189 , 16387128 , 16012269 , 15682680 , 15438250 , 15398936 , 15201368 , 15187041 , 15077907 , 14978061 , 14916727 , 14765192 , 14482503 , 14197248 , 13915125 , 13874112 , 13851000 , 13832000 , 13825728 , 13731319 , 13623913 , 13490533 , 13489875 , 13484016 , 12874176 , 12753160 , 12650337 , 12533824 , 12266856 , 11985003 , 11859211 , 11855727 , 11557000 , 11548089 , 11478584 , 11261376 , 11097567 , 10962432 , 10870119 , 10821896 , 10811017 , 10765603 , 10702783 , 10648512 , 10589696 , 10469816 , 10424232 , 10202696 , 10083528 , 10036845 , 9560896 , 9443761 , 9261729 , 9016488 , 8872487 , 8849601 , 8741824 , 8647128 , 8600256 , 8587000 , 8494577 , 8493039 , 8491392 , 8275176 , 8216000 , 8131968 , 8077888 , 8029000 , 7957504 , 7882245 , 7872536 , 7640128 , 7620661 , 7365384 , 7094269 , 7091712 , 7081984 , 6742008 , 6517000 , 6445504 , 6434883 , 6230504 , 6058747 , 6017193 , 6013000 , 5835375 , 5832729 , 5799339 , 5772403 , 5639816 , 5472152 , 5462424 , 5446359 , 5318677 , 5287707 , 5135104 , 5004125 , 4931101 , 4914000 , 4744376 , 4673088 , 4624776 , 4607064 , 4511808 , 4505949 , 4467528 , 4342914 , 4206592 , 4174281 , 4123000 , 4110848 , 4104000 , 4033503 , 3813992 , 3798613 , 3628233 , 3587409 , 3551112 , 3512808 , 3511872 , 3375008 , 3375001 , 3242197 , 3220776 , 2991816 , 2987712 , 2864288 , 2691451 , 2622104 , 2585375 , 2562112 , 2515968 , 2418271 , 2301299 , 2101248 , 2048391 , 1845649 , 1774656 , 1734264 , 1729000 , 1728216 , 1609272 , 1566728 , 1533357 , 1407672 , 1370304 , 1331064 , 1323712 , 1260441 , 1195112 , 1092728 , 1080891 , 1075032 , 1073375 , 1061424 , 1016496 , 1009736 , 994688 , 984067 , 955016 , 920673 , 886464 , 885248 , 842751 , 805688 , 704977 , 684019 , 593047 , 558441 , 525824 , 515375 , 513856 , 513000 , 443889 , 439101 , 402597 , 373464 , 327763 , 320264 , 314496 , 262656 , 216125 , 216027 , 195841 , 171288 , 165464 , 149389 , 134379 , 110808 , 110656 , 65728 , 64232 , 46683 , 40033 , 39312 , 32832 , 20683 , 13832 , 4104 , 1729 \\right] \\]"
      ],
      "text/plain": [
       "(%o8) [91034307, 90527229, 90091008, 89576767, 88780536, 88129323, 88122125, \n",
       "87952501, 87699456, 87579037, [**, 6, **, 87539319], 87483968, 87029432, \n",
       "86960952, 86575168, 86488136, 86368464, 86316741, 86124824, 85622264, \n",
       "85502375, 85492792, 85188096, 84717568, 84637287, 84266000, 83758528, \n",
       "83393856, 81621568, 80779032, 80668224, 80294760, 80208576, 80102574, \n",
       "79692193, 79598125, 77414967, 77335776, 77129864, 76487168, 75550088, \n",
       "74130875, 74097261, 74093832, 73842713, 73819863, 73369088, 72669177, \n",
       "72131904, 71509312, 71292312, 70979896, 70957971, 70796808, 70449093, \n",
       "69934592, 69805125, 69625969, 69583995, 69177024, 68802048, 68696000, \n",
       "68007673, 67956616, 67944312, 67931136, 67173463, 66763333, 66469429, \n",
       "66201408, 65728000, 65573928, 65293317, 65272753, 65055744, 64623104, \n",
       "64421875, 64232000, 64125000, 63660032, 63057960, 62980288, 62135073, \n",
       "62108397, 61568208, 61555832, 61121024, 60965288, 60698521, 60271939, \n",
       "59913728, 58923072, 58751784, 58438072, 57066633, 56754152, 56733696, \n",
       "56655872, 56388339, 55486125, 55306557, 54887625, 54814509, 54269091, \n",
       "54267003, 53936064, 53283923, 53226297, 52739288, 52324272, 52242463, \n",
       "52136000, 51564032, 51508639, 51479064, 51278929, 51240427, 50324625, \n",
       "49933368, 49849856, 49844032, 49832523, 48469976, 48316662, 48137544, \n",
       "48104000, 47915712, 46825128, 46683000, 46661832, 46394712, 46343059, \n",
       "46179224, 46091997, 45882739, 45440551, 45118528, 44409995, 43997625, \n",
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       "32530680, 32268024, 32114143, 31256000, 30511936, 30388904, 29791125, \n",
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       "13915125, 13874112, 13851000, 13832000, 13825728, 13731319, 13623913, \n",
       "13490533, 13489875, 13484016, 12874176, 12753160, 12650337, 12533824, \n",
       "12266856, 11985003, 11859211, 11855727, 11557000, 11548089, 11478584, \n",
       "11261376, 11097567, 10962432, 10870119, 10821896, 10811017, 10765603, \n",
       "10702783, 10648512, 10589696, 10469816, 10424232, 10202696, 10083528, \n",
       "10036845, 9560896, 9443761, 9261729, 9016488, 8872487, 8849601, 8741824, \n",
       "8647128, 8600256, 8587000, 8494577, 8493039, 8491392, 8275176, 8216000, \n",
       "8131968, 8077888, 8029000, 7957504, 7882245, 7872536, 7640128, 7620661, \n",
       "7365384, 7094269, 7091712, 7081984, 6742008, 6517000, 6445504, 6434883, \n",
       "6230504, 6058747, 6017193, 6013000, 5835375, 5832729, 5799339, 5772403, \n",
       "5639816, 5472152, 5462424, 5446359, 5318677, 5287707, 5135104, 5004125, \n",
       "4931101, 4914000, 4744376, 4673088, 4624776, 4607064, 4511808, 4505949, \n",
       "4467528, 4342914, 4206592, 4174281, 4123000, 4110848, 4104000, 4033503, \n",
       "3813992, 3798613, 3628233, 3587409, 3551112, 3512808, 3511872, 3375008, \n",
       "3375001, 3242197, 3220776, 2991816, 2987712, 2864288, 2691451, 2622104, \n",
       "2585375, 2562112, 2515968, 2418271, 2301299, 2101248, 2048391, 1845649, \n",
       "1774656, 1734264, 1729000, 1728216, 1609272, 1566728, 1533357, 1407672, \n",
       "1370304, 1331064, 1323712, 1260441, 1195112, 1092728, 1080891, 1075032, \n",
       "1073375, 1061424, 1016496, 1009736, 994688, 984067, 955016, 920673, 886464, \n",
       "885248, 842751, 805688, 704977, 684019, 593047, 558441, 525824, 515375, \n",
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      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "apply(append,map(lambda([term],\n",
    "                    block([cf:coeff(term,x^hipow(term,x))],\n",
    "                          if cf>4 then [[\"**\",cf,\"**\",hipow(term,x)]]\n",
    "                          elseif cf>3 then [hipow(term,x)]\n",
    "                          else [])),\n",
    "                 args(TT)));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "このリストの一番小さい数が１７２９であることが確認できます。また１７２９と同様に$450^3+1=91125001$までの数で、２つの３乗数で４通り以上の方法で表すことができる数が全てリストアップされています。さらに87539319については２つの３乗数の和で６通りに表すことができることがわかります。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Copyright 2021 Yasuaki Honda\n",
    "\n",
    "### Computing Ramanujan's taxi number using the generating function $ (\\sum_{n=1}^{\\infty}　x^{n^3})^2 $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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