0132_0231_1023_3102
Counting sequence:
1, 1, 2, 6, 20, 61, 172, 461, 1197, 3047, 7659, 19091, 47304, 116680, 286749, 702524, 1716515, 4183999, 10176457, 24702922, 59857639, 144801640, 349756182, 843615138, 2032142054, 4889153082, 11749470571, 28205893549, 67643670550, 162071417091, 387973059296, 927971114628, 2217814153914, 5296537880771, 12640150292926, 30145417135647, 71847849196187, 171136683207433, 407400734032541, 969307303179069, 2305010867842570, 5478559689757166, 13015210644403947, 30905533270507766, 73355016702652825, 174036255991589941, 412736906049307367, 978447292425094536, 2318668315202928497, 5492663748005848242, 13006977431118974684, 30791030719433512956, 72867139295758066864, 172386708945906514140, 407704085268284113349, 963962184235667851979, 2278519674658039243420, 5384277283645064583349, 12720017354420254118062, 30042582674118785950642, 70938100313374311194704, 167462613756531050749305, 395234951818702940181312, 932599101857402430453473, 2200086140301022848589689, 5189101524135814609514587, 12236439583430945141757631, 28849009825978657212550999, 68002065761771739261394620, 160262151218929666658568836, 377624114096179374700887081, 889632197945596325135455968, 2095488763591027704015665279, 4934989346631736876392819931, 11620238322497562838413661941, 27357323444757010074385043702, 64396710370693988702116417387, 151560823347389826710483206508, 356652318889875539594195417106, 839149884508689745891932049414, 1974112640096804625011779101290, 4643481090990239285423542362606, 10920844001924741952073036785167, 25680936397155820624844055379801, 60382202449953465561900188912530, 141955347329617648477370689256471, 333687933393109060303290477487212, 784289450318123971754954600341136, 1843149307500137010198442339908822, 4331058458475218829471360727541087, 10176021910740693974132456144445234, 23906363692329816079631485384637971, 56156692525880955151398137140400199, 131899457100298073285359106348220125, 309769696876947246425182089774848849, 727428326581331294231741686091690209, 1708038807091615606688522552342288350, 4010166188677861536743412651404398474, 9414233379120643326492283723764292567, 22098679185788456445026605998954880762, 51868923366184658803784283835624083669
Generating function in Maple syntax:
(x^9-6*x^8+17*x^7-33*x^6+51*x^5-59*x^4+50*x^3-27*x^2+8*x-1)/(x^3-2*x^2+3*x-1)^2/(x-1)^3
Generating function in latex syntax:
\frac{x^{9}-6 x^{8}+17 x^{7}-33 x^{6}+51 x^{5}-59 x^{4}+50 x^{3}-27 x^{2}+8 x -1}{\left(x^{3}-2 x^{2}+3 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**9 - 6*x**8 + 17*x**7 - 33*x**6 + 51*x**5 - 59*x**4 + 50*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**3*(x**3 - 2*x**2 + 3*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x^3-2*x^2+3*x-1)^2*(x-1)^3*F(x)-x^9+6*x^8-17*x^7+33*x^6-51*x^5+59*x^4-50*x^3+27*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-2 x^{2}+3 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)-x^{9}+6 x^{8}-17 x^{7}+33 x^{6}-51 x^{5}+59 x^{4}-50 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/317400*(92*2^(1/3)*(((2*n+195/92)*23^(1/2)+I*n-305/4*I)*3^(1/2)+(6*I*n+585/92*I)*23^(1/2)+n-305/4)*(11+3*23^(1/2)*3^(1/2))^(2/3)+3910*2^(2/3)*(((1/17*n-189/391)*23^(1/2)+I*n-16/17*I)*3^(1/2)+(-3/17*I*n+567/391*I)*23^(1/2)-n+16/17)*(11+3*23^(1/2)*3^(1/2))^(1/3)+59800*n-105800)*(11/600*2^(1/3)*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/317400*(-3910*2^(2/3)*(((-1/17*n+189/391)*23^(1/2)+I*n-16/17*I)*3^(1/2)+(-3/17*I*n+567/391*I)*23^(1/2)+n-16/17)*(11+3*23^(1/2)*3^(1/2))^(1/3)-92*(((-2*n-195/92)*23^(1/2)+I*n-305/4*I)*3^(1/2)+(6*I*n+585/92*I)*23^(1/2)-n+305/4)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+59800*n-105800)*(-11/600*2^(1/3)*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/317400*(-460*2^(2/3)*(3^(1/2)*(n-189/23)*23^(1/2)-17*n+16)*(11+3*23^(1/2)*3^(1/2))^(1/3)-368*(3^(1/2)*(n+195/184)*23^(1/2)+1/2*n-305/8)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+59800*n-105800)*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)-1/2*(n+2)*(n-1))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(92 \,2^{\frac{1}{3}} \left(\left(\left(2 n +\frac{195}{92}\right) \sqrt{23}+\mathrm{I} n -\frac{305 \,\mathrm{I}}{4}\right) \sqrt{3}+\left(6 \,\mathrm{I} n +\frac{585 \,\mathrm{I}}{92}\right) \sqrt{23}+n -\frac{305}{4}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+3910 \,2^{\frac{2}{3}} \left(\left(\left(\frac{n}{17}-\frac{189}{391}\right) \sqrt{23}+\mathrm{I} n -\frac{16 \,\mathrm{I}}{17}\right) \sqrt{3}+\left(-\frac{3 \,\mathrm{I} n}{17}+\frac{567 \,\mathrm{I}}{391}\right) \sqrt{23}-n +\frac{16}{17}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+59800 n -105800\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\+\\\frac{\left(-3910 \,2^{\frac{2}{3}} \left(\left(\left(-\frac{n}{17}+\frac{189}{391}\right) \sqrt{23}+\mathrm{I} n -\frac{16 \,\mathrm{I}}{17}\right) \sqrt{3}+\left(-\frac{3 \,\mathrm{I} n}{17}+\frac{567 \,\mathrm{I}}{391}\right) \sqrt{23}+n -\frac{16}{17}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-92 \left(\left(\left(-2 n -\frac{195}{92}\right) \sqrt{23}+\mathrm{I} n -\frac{305 \,\mathrm{I}}{4}\right) \sqrt{3}+\left(6 \,\mathrm{I} n +\frac{585 \,\mathrm{I}}{92}\right) \sqrt{23}-n +\frac{305}{4}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+59800 n -105800\right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\+\\\frac{\left(-460 \,2^{\frac{2}{3}} \left(\sqrt{3}\, \left(n -\frac{189}{23}\right) \sqrt{23}-17 n +16\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-368 \left(\sqrt{3}\, \left(n +\frac{195}{184}\right) \sqrt{23}+\frac{n}{2}-\frac{305}{8}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+59800 n -105800\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{317400}\\-\frac{\left(n +2\right) \left(n -1\right)}{2} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 61
a(6) = 172
a(7) = 461
a(8) = 1197
a(9) = 3047
a(n+6) = -a(n)+4*a(n+1)-10*a(n+2)+14*a(n+3)-13*a(n+4)+6*a(n+5)-1/2*(n+4)*(n+1), n >= 10
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 20
a \! \left(5\right) = 61
a \! \left(6\right) = 172
a \! \left(7\right) = 461
a \! \left(8\right) = 1197
a \! \left(9\right) = 3047
a \! \left(n +6\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right)-\frac{\left(n +4\right) \left(n +1\right)}{2}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15328/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[25,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[18,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[22,x]+F[4,x]
F[22,x] = F[23,x]
F[23,x] = F[21,x]*F[4,x]
F[24,x] = F[19,x]
F[25,x] = F[26,x]+F[38,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[1,x]+F[14,x]
F[30,x] = F[26,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[26,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[34,x]*F[4,x]
F[37,x] = F[31,x]
F[38,x] = F[39,x]+F[40,x]+F[93,x]
F[39,x] = 0
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[52,x]
F[42,x] = F[43,x]+F[46,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[1,x]+F[43,x]
F[46,x] = F[19,x]+F[39,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[14,x]+F[49,x]
F[49,x] = F[39,x]+F[47,x]+F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[21,x]
F[52,x] = F[53,x]+F[74,x]
F[53,x] = F[39,x]+F[40,x]+F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[26,x]+F[56,x]
F[56,x] = F[39,x]+F[54,x]+F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[43,x]+F[49,x]
F[60,x] = F[61,x]+F[64,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[26,x]+F[61,x]
F[64,x] = 2*F[39,x]+F[65,x]+F[72,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[68,x]+F[70,x]
F[68,x] = F[69,x]
F[69,x] = F[26,x]*F[4,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[67,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[31,x]+F[64,x]
F[74,x] = 2*F[39,x]+F[75,x]+F[90,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[77,x]+F[89,x]
F[77,x] = F[78,x]+F[87,x]
F[78,x] = F[39,x]+F[69,x]+F[79,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[81,x]+F[84,x]
F[81,x] = F[4,x]+F[82,x]
F[82,x] = F[19,x]+F[39,x]+F[83,x]
F[83,x] = F[14,x]*F[4,x]
F[84,x] = F[78,x]+F[85,x]
F[85,x] = 2*F[39,x]+F[75,x]+F[86,x]
F[86,x] = F[31,x]*F[4,x]
F[87,x] = F[88,x]
F[88,x] = F[4,x]*F[77,x]
F[89,x] = F[75,x]
F[90,x] = F[4,x]*F[91,x]
F[91,x] = F[31,x]+F[92,x]
F[92,x] = 2*F[39,x]+F[65,x]+F[90,x]
F[93,x] = F[4,x]*F[94,x]
F[94,x] = F[95,x]+F[98,x]
F[95,x] = F[26,x]+F[96,x]
F[96,x] = F[39,x]+F[79,x]+F[97,x]
F[97,x] = F[4,x]*F[95,x]
F[98,x] = F[103,x]+F[99,x]
F[99,x] = F[100,x]+F[39,x]+F[57,x]
F[100,x] = F[101,x]*F[4,x]
F[101,x] = F[102,x]+F[34,x]
F[102,x] = F[99,x]
F[103,x] = F[104,x]
F[104,x] = F[105,x]*F[4,x]
F[105,x] = F[106,x]+F[77,x]
F[106,x] = F[104,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{25}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{38}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{26}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{31}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)+F_{93}\! \left(x \right)
F_{39}\! \left(x \right) = 0
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{52}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{1}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{19}\! \left(x \right)+F_{39}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{39}\! \left(x \right)+F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{21}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{74}\! \left(x \right)
F_{53}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{26}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{39}\! \left(x \right)+F_{54}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{43}\! \left(x \right)+F_{49}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{64}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{26}\! \left(x \right)+F_{61}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{39}\! \left(x \right)+F_{65}\! \left(x \right)+F_{72}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{31}\! \left(x \right)+F_{64}\! \left(x \right)
F_{74}\! \left(x \right) = 2 F_{39}\! \left(x \right)+F_{75}\! \left(x \right)+F_{90}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{89}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{87}\! \left(x \right)
F_{78}\! \left(x \right) = F_{39}\! \left(x \right)+F_{69}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{84}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{19}\! \left(x \right)+F_{39}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{84}\! \left(x \right) = F_{78}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = 2 F_{39}\! \left(x \right)+F_{75}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{89}\! \left(x \right) = F_{75}\! \left(x \right)
F_{90}\! \left(x \right) = F_{4}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{31}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = 2 F_{39}\! \left(x \right)+F_{65}\! \left(x \right)+F_{90}\! \left(x \right)
F_{93}\! \left(x \right) = F_{4}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)+F_{98}\! \left(x \right)
F_{95}\! \left(x \right) = F_{26}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{39}\! \left(x \right)+F_{79}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{4}\! \left(x \right) F_{95}\! \left(x \right)
F_{98}\! \left(x \right) = F_{103}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{39}\! \left(x \right)+F_{57}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{4}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{34}\! \left(x \right)
F_{102}\! \left(x \right) = F_{99}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right) F_{4}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{77}\! \left(x \right)
F_{106}\! \left(x \right) = F_{104}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_25(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_22(x) + F_4(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_21(x)*F_4(x))
Eq(F_24(x), F_19(x))
Eq(F_25(x), F_26(x) + F_38(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_1(x) + F_14(x))
Eq(F_30(x), F_26(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_26(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_34(x)*F_4(x))
Eq(F_37(x), F_31(x))
Eq(F_38(x), F_39(x) + F_40(x) + F_93(x))
Eq(F_39(x), 0)
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_52(x))
Eq(F_42(x), F_43(x) + F_46(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_1(x) + F_43(x))
Eq(F_46(x), F_19(x) + F_39(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_14(x) + F_49(x))
Eq(F_49(x), F_39(x) + F_47(x) + F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_21(x))
Eq(F_52(x), F_53(x) + F_74(x))
Eq(F_53(x), F_39(x) + F_40(x) + F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_26(x) + F_56(x))
Eq(F_56(x), F_39(x) + F_54(x) + F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_43(x) + F_49(x))
Eq(F_60(x), F_61(x) + F_64(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_26(x) + F_61(x))
Eq(F_64(x), 2*F_39(x) + F_65(x) + F_72(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_68(x) + F_70(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_26(x)*F_4(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_67(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_31(x) + F_64(x))
Eq(F_74(x), 2*F_39(x) + F_75(x) + F_90(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_77(x) + F_89(x))
Eq(F_77(x), F_78(x) + F_87(x))
Eq(F_78(x), F_39(x) + F_69(x) + F_79(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_81(x) + F_84(x))
Eq(F_81(x), F_4(x) + F_82(x))
Eq(F_82(x), F_19(x) + F_39(x) + F_83(x))
Eq(F_83(x), F_14(x)*F_4(x))
Eq(F_84(x), F_78(x) + F_85(x))
Eq(F_85(x), 2*F_39(x) + F_75(x) + F_86(x))
Eq(F_86(x), F_31(x)*F_4(x))
Eq(F_87(x), F_88(x))
Eq(F_88(x), F_4(x)*F_77(x))
Eq(F_89(x), F_75(x))
Eq(F_90(x), F_4(x)*F_91(x))
Eq(F_91(x), F_31(x) + F_92(x))
Eq(F_92(x), 2*F_39(x) + F_65(x) + F_90(x))
Eq(F_93(x), F_4(x)*F_94(x))
Eq(F_94(x), F_95(x) + F_98(x))
Eq(F_95(x), F_26(x) + F_96(x))
Eq(F_96(x), F_39(x) + F_79(x) + F_97(x))
Eq(F_97(x), F_4(x)*F_95(x))
Eq(F_98(x), F_103(x) + F_99(x))
Eq(F_99(x), F_100(x) + F_39(x) + F_57(x))
Eq(F_100(x), F_101(x)*F_4(x))
Eq(F_101(x), F_102(x) + F_34(x))
Eq(F_102(x), F_99(x))
Eq(F_103(x), F_104(x))
Eq(F_104(x), F_105(x)*F_4(x))
Eq(F_105(x), F_106(x) + F_77(x))
Eq(F_106(x), F_104(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 0], [1, 0], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], 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