0132_0231_1302_2130
Counting sequence:
1, 1, 2, 6, 20, 65, 202, 608, 1794, 5228, 15112, 43444, 124424, 355424, 1013448, 2886064, 8211552, 23349328, 66364192, 188564480, 535663648, 1521452736, 4320940672, 12270589760, 34844123264, 98941258240, 280940205184, 797705005824, 2264983662080, 6431078921472, 18259958821376, 51845822468096, 147206283518464, 417963104021504, 1186721611302912, 3369451964871680, 9566858359736320, 27163090276900864, 77123874548074496, 218976939703513088, 621738612974247936, 1765294785776472064, 5012179257810690048, 14231016488474902528, 40405941242988863488, 114724063031235362816, 325734531125052080128, 924853776851477151744, 2625925152948636123136, 7455754654599670267904, 21169025736228951326720, 60104934984258117369856, 170655147005003446484992, 484538900962814102667264, 1375744888614508626575360, 3906134250246284522815488, 11090635259711730858721280, 31489493837608800733626368, 89407702857116844818956288, 253854106687527630606172160, 720764603177004729710936064, 2046457392032430056246607872, 5810479369517332507734310912, 16497617117734771146131243008, 46841465779816918427598061568, 132996353393127861664370130944, 377614784700242530285419233280, 1072156656811150941431529472000, 3044160195276006228097455095808, 8643243723352139927603088719872, 24540647426111304121622688956416, 69677935201957604113747502170112, 197835638550827349586653812359168, 561712108249177329660767382798336, 1594861749200266607575742814355456, 4528269840911914768936796407463936, 12857056583356753220972923251064832, 36504870468203498855710341229182976, 103647795221258035741651961179537408, 294285812179026101315369719207821312, 835561808767668154665606500380049408, 2372399576797462074202880618665082880, 6735922696477983459417095054789967872, 19125216096258933007462589031728021504, 54301972752721744655136773718019145728, 154178872018813258138629002000617963520, 437758029256896009207090237485598375936, 1242920574457798672100510781959454064640, 3529007924841285524883724767063646928896, 10019865460047836143612605585721944702976, 28449271289741993247247683179220052213760, 80775639168459586320570036589455629877248, 229345202434966818293039178175058012012544, 651176795645514461809421039426193256873984, 1848877651179112841556765731634139487010816, 5249493827004333345275653552083275440193536, 14904818294591923317894786214451679317721088, 42319053172708610332390847820347392466616320, 120155927166475464266137124942960399440936960, 341157132564251361405977823749777378462662656, 968642927936521007294550776528806684880011264
Generating function in Maple syntax:
(x^5+2*x^4-4*x^3+8*x^2-5*x+1)/(2*x-1)/(2*x^3-4*x^2+4*x-1)
Generating function in latex syntax:
\frac{x^{5}+2 x^{4}-4 x^{3}+8 x^{2}-5 x +1}{\left(2 x -1\right) \left(2 x^{3}-4 x^{2}+4 x -1\right)}
Generating function in sympy syntax:
(x**5 + 2*x**4 - 4*x**3 + 8*x**2 - 5*x + 1)/((2*x - 1)*(2*x**3 - 4*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(2*x^3-4*x^2+4*x-1)*F(x)+x^5+2*x^4-4*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(2 x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)+x^{5}+2 x^{4}-4 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/4224*(-286*((I+17/143*11^(1/2))*3^(1/2)-51/143*I*11^(1/2)-1)*2^(1/3)*(13+3*11^(1/2)*3^(1/2))^(1/3)+704+11*2^(2/3)*((I-13/11*11^(1/2))*3^(1/2)-39/11*I*11^(1/2)+1)*(13+3*11^(1/2)*3^(1/2))^(2/3))*(13/384*2^(2/3)*((I-3/13*11^(1/2))*3^(1/2)-9/13*I*11^(1/2)+1)*(13+3*11^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(26+6*11^(1/2)*3^(1/2))^(1/3)+1/12*(26+6*11^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/4224*(286*2^(1/3)*((I-17/143*11^(1/2))*3^(1/2)-51/143*I*11^(1/2)+1)*(13+3*11^(1/2)*3^(1/2))^(1/3)+704-11*2^(2/3)*((I+13/11*11^(1/2))*3^(1/2)-39/11*I*11^(1/2)-1)*(13+3*11^(1/2)*3^(1/2))^(2/3))*(-13/384*2^(2/3)*((I+3/13*11^(1/2))*3^(1/2)-9/13*I*11^(1/2)-1)*(13+3*11^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(26+6*11^(1/2)*3^(1/2))^(1/3)+1/12*(26+6*11^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/4224*((68*2^(1/3)*11^(1/2)*3^(1/2)-572*2^(1/3))*(13+3*11^(1/2)*3^(1/2))^(1/3)+704+(26*2^(2/3)*11^(1/2)*3^(1/2)-22*2^(2/3))*(13+3*11^(1/2)*3^(1/2))^(2/3))*(1/192*2^(2/3)*(3*11^(1/2)*3^(1/2)-13)*(13+3*11^(1/2)*3^(1/2))^(2/3)-1/6*(26+6*11^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)-5/8*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(-286 \left(\left(\mathrm{I}+\frac{17 \sqrt{11}}{143}\right) \sqrt{3}-\frac{51 \,\mathrm{I} \sqrt{11}}{143}-1\right) 2^{\frac{1}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+704+11 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{13 \sqrt{11}}{11}\right) \sqrt{3}-\frac{39 \,\mathrm{I} \sqrt{11}}{11}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{13 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{11}}{13}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{11}}{13}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}\, \left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{4224}\\+\\\frac{\left(286 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{17 \sqrt{11}}{143}\right) \sqrt{3}-\frac{51 \,\mathrm{I} \sqrt{11}}{143}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+704-11 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{13 \sqrt{11}}{11}\right) \sqrt{3}-\frac{39 \,\mathrm{I} \sqrt{11}}{11}-1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{13 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{11}}{13}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{11}}{13}-1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}+\frac{\mathrm{I} \sqrt{3}\, \left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{4224}\\+\\\frac{\left(\left(68 \,2^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-572 \,2^{\frac{1}{3}}\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+704+\left(26 \,2^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}-22 \,2^{\frac{2}{3}}\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{2}{3}} \left(3 \sqrt{11}\, \sqrt{3}-13\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{192}-\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{4224}\\-\frac{5 \,2^{n}}{8} & \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) = 65
a(n+4) = -4*a(n)+10*a(n+1)-12*a(n+2)+6*a(n+3), n >= 6
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) = 65
a \! \left(n +4\right) = -4 a \! \left(n \right)+10 a \! \left(n +1\right)-12 a \! \left(n +2\right)+6 a \! \left(n +3\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15359/
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[12,x]*F[4,x]
F[4,x] = F[22,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[23,x]+F[46,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[33,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[9,x]
F[30,x] = F[14,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[30,x]
F[33,x] = F[23,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[23,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[12,x]*F[37,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[23,x]+F[41,x]
F[44,x] = F[45,x]
F[45,x] = F[12,x]*F[40,x]
F[46,x] = F[18,x]+F[47,x]+F[84,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[49,x]+F[62,x]
F[49,x] = F[50,x]+F[6,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[10,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[53,x]
F[56,x] = F[57,x]+F[60,x]
F[57,x] = F[58,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[10,x]+F[57,x]
F[60,x] = F[61,x]
F[61,x] = F[12,x]*F[56,x]
F[62,x] = F[46,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[66,x]+F[78,x]
F[66,x] = F[67,x]+F[76,x]
F[67,x] = F[18,x]+F[68,x]+F[74,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[70,x]+F[72,x]
F[70,x] = F[10,x]+F[71,x]
F[71,x] = F[51,x]
F[72,x] = F[67,x]+F[73,x]
F[73,x] = F[64,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[23,x]+F[67,x]
F[76,x] = F[77,x]
F[77,x] = F[12,x]*F[66,x]
F[78,x] = F[79,x]+F[82,x]
F[79,x] = F[80,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[67,x]+F[79,x]
F[82,x] = F[83,x]
F[83,x] = F[12,x]*F[78,x]
F[84,x] = F[12,x]*F[85,x]
F[85,x] = F[75,x]+F[86,x]
F[86,x] = F[108,x]+F[87,x]
F[87,x] = F[106,x]+F[18,x]+F[88,x]
F[88,x] = F[12,x]*F[89,x]
F[89,x] = F[90,x]+F[97,x]
F[90,x] = F[14,x]+F[91,x]
F[91,x] = F[92,x]
F[92,x] = F[12,x]*F[93,x]
F[93,x] = F[94,x]
F[94,x] = F[12,x]+F[95,x]
F[95,x] = F[96,x]
F[96,x] = F[12,x]*F[94,x]
F[97,x] = F[41,x]+F[98,x]
F[98,x] = F[99,x]
F[99,x] = F[100,x]*F[12,x]
F[100,x] = F[101,x]
F[101,x] = F[102,x]+F[104,x]
F[102,x] = F[103,x]
F[103,x] = F[12,x]*F[23,x]
F[104,x] = F[105,x]
F[105,x] = F[101,x]*F[12,x]
F[106,x] = F[107,x]*F[12,x]
F[107,x] = F[23,x]+F[87,x]
F[108,x] = F[109,x]+F[129,x]+F[131,x]+F[18,x]
F[109,x] = F[110,x]*F[12,x]
F[110,x] = F[111,x]+F[120,x]
F[111,x] = F[112,x]+F[57,x]
F[112,x] = F[113,x]
F[113,x] = F[114,x]*F[12,x]
F[114,x] = F[115,x]
F[115,x] = F[116,x]+F[118,x]
F[116,x] = F[117,x]
F[117,x] = F[10,x]*F[12,x]
F[118,x] = F[119,x]
F[119,x] = F[115,x]*F[12,x]
F[120,x] = F[121,x]+F[79,x]
F[121,x] = F[122,x]
F[122,x] = F[12,x]*F[123,x]
F[123,x] = F[124,x]
F[124,x] = F[125,x]+F[127,x]
F[125,x] = F[126,x]
F[126,x] = F[12,x]*F[67,x]
F[127,x] = F[128,x]
F[128,x] = F[12,x]*F[124,x]
F[129,x] = F[12,x]*F[130,x]
F[130,x] = F[108,x]+F[67,x]
F[131,x] = F[12,x]*F[86,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{22}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{46}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{33}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{9}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{23}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{23}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{23}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{46}\! \left(x \right) = F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{84}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{62}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{6}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{10}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{60}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{10}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{62}\! \left(x \right) = F_{46}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{78}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{76}\! \left(x \right)
F_{67}\! \left(x \right) = F_{18}\! \left(x \right)+F_{68}\! \left(x \right)+F_{74}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{72}\! \left(x \right)
F_{70}\! \left(x \right) = F_{10}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{51}\! \left(x \right)
F_{72}\! \left(x \right) = F_{67}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{64}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{23}\! \left(x \right)+F_{67}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{82}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{67}\! \left(x \right)+F_{79}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{75}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{108}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{106}\! \left(x \right)+F_{18}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{12}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{90}\! \left(x \right)+F_{97}\! \left(x \right)
F_{90}\! \left(x \right) = F_{14}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{12}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{12}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{12}\! \left(x \right) F_{94}\! \left(x \right)
F_{97}\! \left(x \right) = F_{41}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right) F_{12}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{104}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)
F_{103}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)
F_{105}\! \left(x \right) = F_{101}\! \left(x \right) F_{12}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right) F_{12}\! \left(x \right)
F_{107}\! \left(x \right) = F_{23}\! \left(x \right)+F_{87}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{129}\! \left(x \right)+F_{131}\! \left(x \right)+F_{18}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right) F_{12}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{120}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)+F_{57}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right) F_{12}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{118}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)
F_{117}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right)
F_{119}\! \left(x \right) = F_{115}\! \left(x \right) F_{12}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)+F_{79}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)
F_{122}\! \left(x \right) = F_{12}\! \left(x \right) F_{123}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)+F_{127}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)
F_{126}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)
F_{128}\! \left(x \right) = F_{12}\! \left(x \right) F_{124}\! \left(x \right)
F_{129}\! \left(x \right) = F_{12}\! \left(x \right) F_{130}\! \left(x \right)
F_{130}\! \left(x \right) = F_{108}\! \left(x \right)+F_{67}\! \left(x \right)
F_{131}\! \left(x \right) = F_{12}\! \left(x \right) F_{86}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_22(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_23(x) + F_46(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_33(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_9(x))
Eq(F_30(x), F_14(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_30(x))
Eq(F_33(x), F_23(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_23(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_12(x)*F_37(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_23(x) + F_41(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x)*F_40(x))
Eq(F_46(x), F_18(x) + F_47(x) + F_84(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_62(x))
Eq(F_49(x), F_50(x) + F_6(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_10(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_53(x))
Eq(F_56(x), F_57(x) + F_60(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_10(x) + F_57(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_12(x)*F_56(x))
Eq(F_62(x), F_46(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_78(x))
Eq(F_66(x), F_67(x) + F_76(x))
Eq(F_67(x), F_18(x) + F_68(x) + F_74(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_72(x))
Eq(F_70(x), F_10(x) + F_71(x))
Eq(F_71(x), F_51(x))
Eq(F_72(x), F_67(x) + F_73(x))
Eq(F_73(x), F_64(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_23(x) + F_67(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_12(x)*F_66(x))
Eq(F_78(x), F_79(x) + F_82(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_67(x) + F_79(x))
Eq(F_82(x), F_83(x))
Eq(F_83(x), F_12(x)*F_78(x))
Eq(F_84(x), F_12(x)*F_85(x))
Eq(F_85(x), F_75(x) + F_86(x))
Eq(F_86(x), F_108(x) + F_87(x))
Eq(F_87(x), F_106(x) + F_18(x) + F_88(x))
Eq(F_88(x), F_12(x)*F_89(x))
Eq(F_89(x), F_90(x) + F_97(x))
Eq(F_90(x), F_14(x) + F_91(x))
Eq(F_91(x), F_92(x))
Eq(F_92(x), F_12(x)*F_93(x))
Eq(F_93(x), F_94(x))
Eq(F_94(x), F_12(x) + F_95(x))
Eq(F_95(x), F_96(x))
Eq(F_96(x), F_12(x)*F_94(x))
Eq(F_97(x), F_41(x) + F_98(x))
Eq(F_98(x), F_99(x))
Eq(F_99(x), F_100(x)*F_12(x))
Eq(F_100(x), F_101(x))
Eq(F_101(x), F_102(x) + F_104(x))
Eq(F_102(x), F_103(x))
Eq(F_103(x), F_12(x)*F_23(x))
Eq(F_104(x), F_105(x))
Eq(F_105(x), F_101(x)*F_12(x))
Eq(F_106(x), F_107(x)*F_12(x))
Eq(F_107(x), F_23(x) + F_87(x))
Eq(F_108(x), F_109(x) + F_129(x) + F_131(x) + F_18(x))
Eq(F_109(x), F_110(x)*F_12(x))
Eq(F_110(x), F_111(x) + F_120(x))
Eq(F_111(x), F_112(x) + F_57(x))
Eq(F_112(x), F_113(x))
Eq(F_113(x), F_114(x)*F_12(x))
Eq(F_114(x), F_115(x))
Eq(F_115(x), F_116(x) + F_118(x))
Eq(F_116(x), F_117(x))
Eq(F_117(x), F_10(x)*F_12(x))
Eq(F_118(x), F_119(x))
Eq(F_119(x), F_115(x)*F_12(x))
Eq(F_120(x), F_121(x) + F_79(x))
Eq(F_121(x), F_122(x))
Eq(F_122(x), F_12(x)*F_123(x))
Eq(F_123(x), F_124(x))
Eq(F_124(x), F_125(x) + F_127(x))
Eq(F_125(x), F_126(x))
Eq(F_126(x), F_12(x)*F_67(x))
Eq(F_127(x), F_128(x))
Eq(F_128(x), F_12(x)*F_124(x))
Eq(F_129(x), F_12(x)*F_130(x))
Eq(F_130(x), F_108(x) + F_67(x))
Eq(F_131(x), F_12(x)*F_86(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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "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": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, 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