A figurate_numbers.txt f8adb7a0367d0442d19431694bc7f7ef38eb36f5 1d6cd015562e181feb6759bb3db167ad575b0606419d9d6d802b8071224b27ba
C text\sfiles\r\n\r\nf8adb7a0367d0442d19431694bc7f7ef38eb36f5\r\nfigurate\snumbers\sand\ssums\sof\spowers\r\n\r\n\r\nprograms\salso\sloaded\son\sTCL\sWIKI\spages.\r\nAlso\sOEIS\shas\scontent\son\sfigurate\snumbers.\r\nsuggest\sfigurate\snumbers\sand\ssums\sof\spowers\s(formulas)\sinto\sTCLLIB\smath\smodule.\r\n\r\n(body)\sSuggest\sload\sfigurate\snumbers\sand\ssums\sof\spowers\s(formulas)\sinto\sTCLLIB\s\smath\smodule..\sI\sloaded\ssome\sone\sliner\sprograms\sand\sformulas\s\son\sthe\sTCL\sWIKI\spages.\sAlso\sOEIS\shas\ssubstantial\s\scontent\son\sfigurate\snumbers.\r\n\r\nhttps://oeis.org/search?q=figurate&sort=&language=english&go=Search\r\nhttps://en.wikipedia.org/wiki/Figurate_number\r\nhttps://wiki.tcl-lang.org/page/One+Liners+Programs+Compendium++and+TCL+demo+examples+calculations%2C+numerical+analysis\r\nhttps://wiki.tcl-lang.org/page/Triangular+Number+Multiplication+Study++and++demo+example+TCL+calculator%2C+numerical+analysis\r\n\r\n\r\nOne\sLiners\sPrograms\sCompendium\sand\sTCL\sdemo\sexamples\scalculations,\snumerical\sanalysis\r\n\r\n----\r\n***\sAdvanced\sTopics:\sFigurate\sPolynomial\sFormulas\sfor\sTCL\sProcs\s\s\s***\r\n----\r\n----\r\n***\sTable\s\s:\s\sFigurate\sFormulas\sand\sTCL\sProcs\sTable\s\s***\r\n----\r\n%|Table\sFigurate\sFormulas\sinto\sTCL\sProcs\s\s||\s\s\s\s|printed\sin\s|\stcl\sformat|%\s\r\n%|\s\s\s\s\s\s\s\sfigurate\sname\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sformula\sin\svariable\sn\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sautomatic\strial\s\sTCL\s\sproc\s************************\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sinfinite\sseries\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\scomment,\sif\sany\s\s\s\s\s\s\s\s|%\r\n&|\s\s\s\s\s\s\s\soblong\snumbers\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sn*(n+1)\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sproc\sfigurate1\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{($n+1)\s}]}\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\s0,\s2,\s6,\s12,\s20,\s30,\s42,\s56,\s72,\s90,\s110,\s...\s\s\s\s\s\s\s\s\s|\s|&\r\n&|\s\s\s\s\s\s\s\striangular\snumbers\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sn*(n+1)/2\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sproc\sfigurate1\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{($n+1)/2}]}\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\s1\s3\s6\s10\s15\s21\s28...\s\s\s\s\s\s\s\s\s|\s|&\r\n&|\s\s\s\s\s\s\s\ssquare\snumbers\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sn**2\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sproc\sfigurate1\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{$n**2}]}\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\s1\s4\s9\s16\s25\s36\s49...\s\s\s\s\s\s\s\s\s|\s|&\r\n&|\s\s\s\s\s\s\s\spentagonal\snumbers\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sn*(3n-1)/2\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sproc\sfigurate1\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{(3*$n-1)/2}]}\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\s1\s5\s12\s22\s35\s51\s70...\s\s\s\s\s\s\s\s\s|\s|&\r\n&|\s\s\s\s\s\s\s\shexagonal\snumbers\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sn*(4n-2)/2\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sproc\sfigurate1\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{(4*$n-2)/2}]}\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\s1\s6\s15\s28\s45\s66\s91...\s\s\s\s\s\s\s\s\s|\s|&\r\n&|\s\s\s\s\s\s\s\sheptagonal\snumbers\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sn*(5n-3)/2\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sproc\sfigurate1\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{(5*$n-3)/2}]}\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\s1\s7\s18\s34\s55\s81\s112...\s\s\s\s\s\s\s\s\s|\s|&\r\n&|\s\s\s\s\s\s\s\soctogonal\snumbers\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sn*(3n-2)\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sproc\sfigurate1\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{(3*$n-2)}]}\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\s1\s8\s21\s40\s65\s96\s133...\s\s\s\s\s\s\s\s\s|\s|&\r\n&|\s\s\s\s\s\s\s\s**********************\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\s\s2D\sfigurate\snumbers\sinto\s3D\sfigurate\snumbers\s\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\s************************\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s|\s|&\r\n&|\s\s\s\s\s\s\s\striangular\snumbers\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sn*(n+1)/2\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sproc\sfigurate1\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{($n+1)/2}]}\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\s1\s3\s6\s10\s15\s21...\s\s\s\s\s\s\s\s\s|\s|&\r\n&|\s\s\s\s\s\s\s\stetrahedral\snumbers\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sn*(n+1)*(n+2)/6\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sproc\sfigurate1\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{($n+1)*($n+2)/6}]}\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\s1\s4\s10\s20\s35\s56...\s\s\s\s\s\s\s\s\s|\s|&\r\n&|\s\s\s\s\s\s\s\shypertetrahedral\snumbers\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sn*(n+1)*(n+2)*(n+3)/24\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\sproc\sfigurate1\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{($n+1)*($n+2)*($n+3)/24}]}\s\s\s\s\s\s\s\s|\s\s\s\s\s\s\s\s1\s5\s15\s35\s70\s126...\s\s\s\s\s\s\s\s\s|\s|&\r\n----\r\nfrom\sGnumeric\sspreadsheet\r\n----\r\n======\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\r\nproc\sfigurate1_oblong\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{($n+1)\s}]}\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\r\nproc\sfigurate2_triangular\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{($n+1)/2}]}\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\r\nproc\sfigurate3_square\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{$n**2}]}\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\r\nproc\sfigurate4_pentagonal\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{(3*$n-1)/2}]}\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\r\nproc\sfigurate5_hexagonal\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{(4*$n-2)/2}]}\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\r\nproc\sfigurate6_heptagonal\s\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{(5*$n-3)/2}]}\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\r\nproc\sfigurate7_octogonal\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{(3*$n-2)}]}\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\r\n;#\s*****\s\s2D\sfigurate\sinto\s3D\sfigurate\s$numbers\s\s\s\s\s\s\s\s******\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\r\nproc\sfigurate8_triangular\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{($n+1)/2}]}\s\s\s;#\sduplicate\sproc\sfor\scomparison\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\r\nproc\sfigurate9_tetrahedral\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{($n+1)*($n+2)/6}]}\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\s\r\nproc\sfigurate10_hypertetrahedral\s{\s\sn\s}\s{[\sreturn\s[\sexpr\s{($n+1)*($n+2)*($n+3)/24}]}\r\n======\s\s\s\s\s\s\s\s\r\n----\s\r\n----\r\n***\sAdvanced\sTopics:\sPolynomials,\sSums\sof\sPowers,\s\sSymbolic\sDifferentiation\s&\sIntegration\s\s\sin\s\sTcllib\sCalculus***\r\n----\r\n======\r\n;#\sadding\sextra\sand\sredundant\sspaces\sin\sproc\sexpressions\s\r\n;#\sbelow\sfor\swiki\sreadability\r\n;#\srecommend\seach\sproc\sbe\smath\schecked\sby\shand\son\ssmall\snumbers\s(2,3,4)\r\n;#\srecommend\seach\sproc\sbe\schecked\sfor\ssmall,\smedium,\s\sand\slarge\snumbers\r\n;#\sin\sexpected\srange\sof\soperation\r\n;#\sone\smay\sleave\soff\sreturn\scommands\sand\sunnecessary\sspaces\sfor\sbrevity\r\n\r\n;#\sformula\sfor\sSum\sof\sSquares\sis\sk(2)\s=\sn*(n+1)*(2*n+1)/6\r\nproc\ssum_squares\s\s{\sn\s}\s{set\sres\s[expr\s{\s$n*($n\s+\s1)*(2*$n\s+1\s)\s/\s6\s}]}\r\n;#\sUsage\ssum_squares\s2\s->\s\s5\s\r\n;#\scheck\sanswer\sexpr\s2**2\s->\s4,\s4\s+\s1\s=\s5\r\n;#\sUsage\ssum_squares\s1000000\s->\s\s333333833333500000\s\r\n\r\n;#\sformula\sfor\sSum\sof\sCubes\sis\sk(3)\s=\s(n**2)*\s((n\s+\s1)**2)\s/\s4\s\r\nproc\ssum_cubes\s\s{\sn\s}\s{set\sres\s[expr\s{\s($n**2)*\s(($n\s+\s1)**2)\s/\s4\s}]}\r\n;#\sUsage\ssum_cubes\s2\s->\s\s9\r\n;#\scheck\sanswer\sexpr\s2**3\s->\s8,\s8\s+\s1\s=\s9\r\n;#\sUsage\ssum_cubes\s1000000\s->\s\s250000500000250000000000\s\r\n\r\n;#\sformula\ssum_4th_power\sk(4)\s=\sn*\s(n\s+\s1)*\s(2*n\s+\s1\s)*\s(\s3*n**2\s+\s3*n\s-1)/30\r\n;#\susing\sinteger\sarithmetic\sfor\sthe\slong\sdigit\sanswers\r\nproc\s\ssum_4th_power\s{n}\s{\sexpr\s{\s$n*\s($n\s+\s1)*\s(2*$n\s+\s1\s)*\s(3*$n**2\s+\s3*$n\s-1)/30}}\r\n;#\sUsage\ssum_4th_power\s2\s\s->\s17\r\n;#\scheck\sanswer\sexpr\s2**4\s->\s16,\s16\s+\s1\s=\s17\r\n;#\sUsage\ssum_4th_power\s0\s\s->\s0\s\s;#\sUsage\ssum_4th_power\s1\s\s->\s1\s\r\n;#\sUsage\ssum_4th_power\s1000\s\s->\s200500333333300\r\n;#\sUsage\s\ssum_4th_power\s200\s->\s64802666660\r\n;#\sUsage\ssum_4th_power\s1000000000\r\n;#\s\s\s\s\s->\s200000000500000000333333333333333333300000000\r\n\s\r\n;#\sformula\ssum_5th_power\sk(5)\s=\sn*n*\s(n+1)*\s(n+1\s)*\s(\s\s2*n**2\s+\s2*n\s-1)/12\r\nproc\ssum_5th\s{n}\s{return\s[expr\s{($n*$n*\s($n\s+\s1)*($n\s+\s1\s)*(2*($n**2)+2*$n\s-\s1))/12}]\s}\r\n;#\sinternal\s?\sin\sexpr\stesting\sfor\s2\sconditions\r\nproc\ssum_5th_power\s\s{n}\s{expr\s{\s$n\s<\s1?\s0:\s$n\s>\s1\s?\s1\s:\s($n*$n*\s($n\s+\s1)*($n\s+\s1\s)*(2*($n**2)+2*$n\s-\s1))/12}}\r\n;#\sUsage\ssum_5th_power\s0\s\s->\s0\s\s;#\sUsage\ssum_5th_power\s1\s\s->\s1\s\r\n;#\sUsage\ssum_5th_power\s2\s\s->\s33\r\n\r\n;#\sformula\ssum_6th_power\sk(6)\s=(\sn*(n\s+\s1)*(2*n\s+\s1\s)*(3*(n**4)+(6*(n**3))-(3*n)\s+\s1))/42\r\nproc\ssum_6th\s{x}\s{return\s[expr\s{(\s$x*($x\s+\s1)*(2*$x\s+\s1\s)*(3*($x**4)+(6*($x**3))-(3*$x)\s+\s1))/42}]\s}\r\n;#\sUsage\ssum_6th\s2\s->\s65\s\r\n;#\scheck\sanswer\sexpr\s2**6\s=\s64,\s1\s+\s64\s=\s65\s\r\n\r\n;#\sformula\ssum_7th_power\sk(7)\s=(\sn*n*(n\s+\s1)*(n\s+\s1)*(3*(n**4)+(6*(n**3))-n*n-(4*n)\s+\s2))/24\r\nproc\ssum_7th\s{x}\s{return\s[expr\s{(\s$x*$x*($x\s+\s1)*($x\s+\s1)*(3*($x**4)+(6*($x**3))-$x*$x-(4*$x)\s+\s2))/24}]\s}\r\n;#\sUsage\ssum_7th\s2\s->\s129\s\r\n;#\scheck\sanswer\s\sexpr\s2**7\s=\s128,\s1+128\s=\s129\r\n======\r\n----\r\n======\r\n;#\sadding\sextra\sand\sredundant\sspaces\sin\sexpressions\s\sbelow\sfor\swiki\sreadability\r\n;#\srecommend\seach\sproc\sbe\smath\schecked\sby\shand\son\ssmall\snumbers\s(2,3,4)\r\n;#\srecommend\seach\sproc\sbe\schecked\sfor\ssmall,\smedium,\sand\slarge\snumbers\r\n;#\sin\sexpected\srange\sof\soperation\r\n;#\sone\smay\sleave\soff\sreturn\scommands\sand\sunnecessary\sspaces
D 2021-05-07T19:30:19.047
U anonymous
Z 0cec118904494c8def8b08342c5a33aa
