0132_1320_2130
Counting sequence:
1, 1, 2, 6, 21, 75, 260, 869, 2817, 8920, 27745, 85113, 258256, 776717, 2319093, 6882432, 20321017, 59738793, 174960544, 510756501, 1486819229, 4317417984, 12509488161, 36175413601, 104433440896, 301022300573, 866485007957, 2491082722400, 7153751750089, 20523229989113, 58825513697056, 168473360357989, 482142577403917, 1378885365897664, 3941085554868529, 11258031480904145, 32143223466398784, 91731000572885741, 261673686920805509, 746168677402454688, 2126969902773252953, 6061024609349480297, 17266445243482962656, 49174817195175190837, 140015232522378995901, 398574443433419757824, 1134366457450518143553, 3227858567723527065025, 9183317488373997078272, 26122509032137906439997, 74296104453992539999925, 211279939660001083565280, 600753506732323804225513, 1707986997733482858724057, 4855422257089245256290464, 13801532771645310916130565, 39227341802199462637884525, 111484573161162186525410880, 316816542804934550478732881, 900265725132100391252341937, 2558030812663030733538746304, 7268005083877107452120263309, 20649098542519015605488803813, 58663133833945700944508630304, 166651576448218468903532707705, 473407544512349634112755566921, 1344757445720320143279740771424, 3819769098145810685070820805589, 10849653414183232486623442913053, 30816360961542548935874990328192, 87525503088128980445229280695649, 248585970635960249401803375212193, 706005743315754255593382893047424, 2005073455198218172951646907790557, 5694341712508784648225937100247061, 16171437928692015664956879153412960, 45924693089877075690856645529870857, 130417834601182713981466737613712057, 370357672182133582958571936862723616, 1051719296810713718107371720810000293, 2986571618358627242543627699058634957, 8480882422174588994801415418092085952, 24082665733099336364678142123828000625, 68385470461439761991751658007911416721, 194186581776567372659937133926995104576, 551405376802255073861061410800572069421, 1565739323233411712002849852585397374789, 4445952955825559489072613099342510159776, 12624304101408714446343081572601984784281, 35846555674558484632927705825776643034921, 101785290724091710673662214130115113178592, 289015011514715824523319536891184279925365, 820642005333068388118392233362977782546813, 2330157126072193074621642465620221295530496, 6616296203216273628287814226562645958358145, 18786378841582889291691966930276747197434753, 53342054675401796472003207781688823143886336, 151458986828740588684460032545768429511048829, 430050149687827237883986242942264445692579189, 1221074059874036648387796846538544454692547040, 3467079848271671904870444237618812600228868137
Generating function in Maple syntax:
-(3*x^5-14*x^4+21*x^3-18*x^2+7*x-1)/(x-1)/(2*x^3-4*x^2+4*x-1)/(x^2-3*x+1)
Generating function in latex syntax:
-\frac{3 x^{5}-14 x^{4}+21 x^{3}-18 x^{2}+7 x -1}{\left(x -1\right) \left(2 x^{3}-4 x^{2}+4 x -1\right) \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(-3*x**5 + 14*x**4 - 21*x**3 + 18*x**2 - 7*x + 1)/((x - 1)*(x**2 - 3*x + 1)*(2*x**3 - 4*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x^3-4*x^2+4*x-1)*(x^2-3*x+1)*F(x)+3*x^5-14*x^4+21*x^3-18*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x^{3}-4 x^{2}+4 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+3 x^{5}-14 x^{4}+21 x^{3}-18 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
1/10560*(-110*((-63/11*I+21/11*3^(1/2))*11^(1/2)+I*3^(1/2)-1)*2^(1/3)*(13+3*11^(1/2)*3^(1/2))^(1/3)+3520-605*((-45/121*I-15/121*3^(1/2))*11^(1/2)+I*3^(1/2)+1)*2^(2/3)*(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)+1/10560*(110*2^(1/3)*((-63/11*I-21/11*3^(1/2))*11^(1/2)+I*3^(1/2)+1)*(13+3*11^(1/2)*3^(1/2))^(1/3)+3520+605*2^(2/3)*((-45/121*I+15/121*3^(1/2))*11^(1/2)+I*3^(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)+1/10560*((420*2^(1/3)*11^(1/2)*3^(1/2)-220*2^(1/3))*(13+3*11^(1/2)*3^(1/2))^(1/3)+3520+(-150*11^(1/2)*3^(1/2)*2^(2/3)+1210*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)+1/10560*(-4224*5^(1/2)-10560)*(3/2-1/2*5^(1/2))^(-n)+2+1/10560*(4224*5^(1/2)-10560)*(3/2+1/2*5^(1/2))^(-n)
Explicit closed form in latex syntax:
\frac{\left(-110 \left(\left(-\frac{63 \,\mathrm{I}}{11}+\frac{21 \sqrt{3}}{11}\right) \sqrt{11}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+3520-605 \left(\left(-\frac{45 \,\mathrm{I}}{121}-\frac{15 \sqrt{3}}{121}\right) \sqrt{11}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{2}{3}} \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}}{10560}+\frac{\left(110 \,2^{\frac{1}{3}} \left(\left(-\frac{63 \,\mathrm{I}}{11}-\frac{21 \sqrt{3}}{11}\right) \sqrt{11}+\mathrm{I} \sqrt{3}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+3520+605 \,2^{\frac{2}{3}} \left(\left(-\frac{45 \,\mathrm{I}}{121}+\frac{15 \sqrt{3}}{121}\right) \sqrt{11}+\mathrm{I} \sqrt{3}-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}}{10560}+\frac{\left(\left(420 \,2^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-220 \,2^{\frac{1}{3}}\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+3520+\left(-150 \sqrt{11}\, \sqrt{3}\, 2^{\frac{2}{3}}+1210 \,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}}{10560}+\frac{\left(-4224 \sqrt{5}-10560\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10560}+2+\frac{\left(4224 \sqrt{5}-10560\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10560}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 75
a(n+5) = 2*a(n)-10*a(n+1)+18*a(n+2)-17*a(n+3)+7*a(n+4)+2, 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) = 21
a \! \left(5\right) = 75
a \! \left(n +5\right) = 2 a \! \left(n \right)-10 a \! \left(n +1\right)+18 a \! \left(n +2\right)-17 a \! \left(n +3\right)+7 a \! \left(n +4\right)+2, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14736/
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[31,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[20,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[17,x]
F[20,x] = F[21,x]+F[22,x]+F[30,x]
F[21,x] = 0
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[24,x]
F[27,x] = F[20,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[27,x]
F[30,x] = F[12,x]*F[13,x]
F[31,x] = F[14,x]+F[32,x]
F[32,x] = F[108,x]+F[21,x]+F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[65,x]
F[35,x] = F[36,x]+F[6,x]
F[36,x] = F[21,x]+F[26,x]+F[37,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[10,x]+F[40,x]
F[40,x] = F[21,x]+F[26,x]+F[41,x]
F[41,x] = F[12,x]*F[39,x]
F[42,x] = F[43,x]+F[52,x]
F[43,x] = F[21,x]+F[26,x]+F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[10,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[46,x]
F[49,x] = F[43,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[49,x]
F[52,x] = F[21,x]+F[53,x]+F[56,x]+F[64,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[25,x]+F[55,x]
F[55,x] = F[53,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = F[40,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[12,x]*F[58,x]
F[61,x] = F[52,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[12,x]*F[61,x]
F[64,x] = F[12,x]*F[42,x]
F[65,x] = F[32,x]+F[66,x]
F[66,x] = 2*F[21,x]+F[67,x]+F[80,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[69,x]+F[79,x]
F[69,x] = F[21,x]+F[70,x]+F[77,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[72,x]+F[73,x]
F[72,x] = F[10,x]+F[40,x]
F[73,x] = F[69,x]+F[74,x]
F[74,x] = 2*F[21,x]+F[67,x]+F[75,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[18,x]+F[74,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[14,x]+F[69,x]
F[79,x] = F[67,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[76,x]+F[82,x]
F[82,x] = F[83,x]+F[95,x]
F[83,x] = 2*F[21,x]+F[84,x]+F[87,x]
F[84,x] = F[12,x]*F[85,x]
F[85,x] = F[18,x]+F[86,x]
F[86,x] = F[84,x]
F[87,x] = F[12,x]*F[88,x]
F[88,x] = F[89,x]+F[92,x]
F[89,x] = F[18,x]+F[90,x]
F[90,x] = F[91,x]
F[91,x] = F[12,x]*F[89,x]
F[92,x] = F[83,x]+F[93,x]
F[93,x] = F[94,x]
F[94,x] = F[12,x]*F[92,x]
F[95,x] = 2*F[21,x]+F[107,x]+F[96,x]+F[99,x]
F[96,x] = F[12,x]*F[97,x]
F[97,x] = F[79,x]+F[98,x]
F[98,x] = F[96,x]
F[99,x] = F[100,x]*F[12,x]
F[100,x] = F[101,x]+F[104,x]
F[101,x] = F[102,x]+F[74,x]
F[102,x] = F[103,x]
F[103,x] = F[101,x]*F[12,x]
F[104,x] = F[105,x]+F[95,x]
F[105,x] = F[106,x]
F[106,x] = F[104,x]*F[12,x]
F[107,x] = F[12,x]*F[82,x]
F[108,x] = F[109,x]*F[12,x]
F[109,x] = F[110,x]+F[78,x]
F[110,x] = F[111,x]+F[119,x]
F[111,x] = F[112,x]+F[114,x]+F[21,x]
F[112,x] = F[113,x]*F[12,x]
F[113,x] = F[24,x]+F[85,x]
F[114,x] = F[115,x]*F[12,x]
F[115,x] = F[116,x]+F[17,x]
F[116,x] = F[111,x]+F[117,x]
F[117,x] = F[118,x]
F[118,x] = F[116,x]*F[12,x]
F[119,x] = F[120,x]+F[130,x]+F[135,x]+F[21,x]
F[120,x] = F[12,x]*F[121,x]
F[121,x] = F[122,x]+F[126,x]
F[122,x] = F[123,x]+F[40,x]
F[123,x] = 2*F[21,x]+F[124,x]+F[53,x]
F[124,x] = F[12,x]*F[125,x]
F[125,x] = F[123,x]+F[25,x]
F[126,x] = F[127,x]+F[74,x]
F[127,x] = 3*F[21,x]+F[128,x]+F[96,x]
F[128,x] = F[12,x]*F[129,x]
F[129,x] = F[127,x]+F[86,x]
F[130,x] = F[12,x]*F[131,x]
F[131,x] = F[132,x]+F[68,x]
F[132,x] = F[119,x]+F[133,x]
F[133,x] = F[134,x]
F[134,x] = F[12,x]*F[132,x]
F[135,x] = F[110,x]*F[12,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_{31}\! \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_{20}\! \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_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{30}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{108}\! \left(x \right)+F_{21}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{65}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{6}\! \left(x \right)
F_{36}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{52}\! \left(x \right)
F_{43}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{43}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{21}\! \left(x \right)+F_{53}\! \left(x \right)+F_{56}\! \left(x \right)+F_{64}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{25}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{61}\! \left(x \right)
F_{58}\! \left(x \right) = F_{40}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{52}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{65}\! \left(x \right) = F_{32}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{67}\! \left(x \right)+F_{80}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{79}\! \left(x \right)
F_{69}\! \left(x \right) = F_{21}\! \left(x \right)+F_{70}\! \left(x \right)+F_{77}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{73}\! \left(x \right)
F_{72}\! \left(x \right) = F_{10}\! \left(x \right)+F_{40}\! \left(x \right)
F_{73}\! \left(x \right) = F_{69}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{67}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{18}\! \left(x \right)+F_{74}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{14}\! \left(x \right)+F_{69}\! \left(x \right)
F_{79}\! \left(x \right) = F_{67}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{76}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{95}\! \left(x \right)
F_{83}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{84}\! \left(x \right)+F_{87}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{18}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{84}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{92}\! \left(x \right)
F_{89}\! \left(x \right) = F_{18}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{12}\! \left(x \right) F_{89}\! \left(x \right)
F_{92}\! \left(x \right) = F_{83}\! \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_{92}\! \left(x \right)
F_{95}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{107}\! \left(x \right)+F_{96}\! \left(x \right)+F_{99}\! \left(x \right)
F_{96}\! \left(x \right) = F_{12}\! \left(x \right) F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{79}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{96}\! \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_{104}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{74}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)
F_{103}\! \left(x \right) = F_{101}\! \left(x \right) F_{12}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{95}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)
F_{106}\! \left(x \right) = F_{104}\! \left(x \right) F_{12}\! \left(x \right)
F_{107}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{12}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)+F_{78}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{119}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)+F_{114}\! \left(x \right)+F_{21}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{12}\! \left(x \right)
F_{113}\! \left(x \right) = F_{24}\! \left(x \right)+F_{85}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right) F_{12}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{17}\! \left(x \right)
F_{116}\! \left(x \right) = F_{111}\! \left(x \right)+F_{117}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)
F_{118}\! \left(x \right) = F_{116}\! \left(x \right) F_{12}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)+F_{130}\! \left(x \right)+F_{135}\! \left(x \right)+F_{21}\! \left(x \right)
F_{120}\! \left(x \right) = F_{12}\! \left(x \right) F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)+F_{126}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)+F_{40}\! \left(x \right)
F_{123}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{124}\! \left(x \right)+F_{53}\! \left(x \right)
F_{124}\! \left(x \right) = F_{12}\! \left(x \right) F_{125}\! \left(x \right)
F_{125}\! \left(x \right) = F_{123}\! \left(x \right)+F_{25}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)+F_{74}\! \left(x \right)
F_{127}\! \left(x \right) = 3 F_{21}\! \left(x \right)+F_{128}\! \left(x \right)+F_{96}\! \left(x \right)
F_{128}\! \left(x \right) = F_{12}\! \left(x \right) F_{129}\! \left(x \right)
F_{129}\! \left(x \right) = F_{127}\! \left(x \right)+F_{86}\! \left(x \right)
F_{130}\! \left(x \right) = F_{12}\! \left(x \right) F_{131}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)+F_{68}\! \left(x \right)
F_{132}\! \left(x \right) = F_{119}\! \left(x \right)+F_{133}\! \left(x \right)
F_{133}\! \left(x \right) = F_{134}\! \left(x \right)
F_{134}\! \left(x \right) = F_{12}\! \left(x \right) F_{132}\! \left(x \right)
F_{135}\! \left(x \right) = F_{110}\! \left(x \right) F_{12}\! \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_31(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_20(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_17(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_30(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_24(x))
Eq(F_27(x), F_20(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_27(x))
Eq(F_30(x), F_12(x)*F_13(x))
Eq(F_31(x), F_14(x) + F_32(x))
Eq(F_32(x), F_108(x) + F_21(x) + F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_65(x))
Eq(F_35(x), F_36(x) + F_6(x))
Eq(F_36(x), F_21(x) + F_26(x) + F_37(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_10(x) + F_40(x))
Eq(F_40(x), F_21(x) + F_26(x) + F_41(x))
Eq(F_41(x), F_12(x)*F_39(x))
Eq(F_42(x), F_43(x) + F_52(x))
Eq(F_43(x), F_21(x) + F_26(x) + F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_10(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_46(x))
Eq(F_49(x), F_43(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_49(x))
Eq(F_52(x), F_21(x) + F_53(x) + F_56(x) + F_64(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_25(x) + F_55(x))
Eq(F_55(x), F_53(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), F_40(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_12(x)*F_58(x))
Eq(F_61(x), F_52(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_12(x)*F_61(x))
Eq(F_64(x), F_12(x)*F_42(x))
Eq(F_65(x), F_32(x) + F_66(x))
Eq(F_66(x), 2*F_21(x) + F_67(x) + F_80(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_79(x))
Eq(F_69(x), F_21(x) + F_70(x) + F_77(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_73(x))
Eq(F_72(x), F_10(x) + F_40(x))
Eq(F_73(x), F_69(x) + F_74(x))
Eq(F_74(x), 2*F_21(x) + F_67(x) + F_75(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_18(x) + F_74(x))
Eq(F_77(x), F_12(x)*F_78(x))
Eq(F_78(x), F_14(x) + F_69(x))
Eq(F_79(x), F_67(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_76(x) + F_82(x))
Eq(F_82(x), F_83(x) + F_95(x))
Eq(F_83(x), 2*F_21(x) + F_84(x) + F_87(x))
Eq(F_84(x), F_12(x)*F_85(x))
Eq(F_85(x), F_18(x) + F_86(x))
Eq(F_86(x), F_84(x))
Eq(F_87(x), F_12(x)*F_88(x))
Eq(F_88(x), F_89(x) + F_92(x))
Eq(F_89(x), F_18(x) + F_90(x))
Eq(F_90(x), F_91(x))
Eq(F_91(x), F_12(x)*F_89(x))
Eq(F_92(x), F_83(x) + F_93(x))
Eq(F_93(x), F_94(x))
Eq(F_94(x), F_12(x)*F_92(x))
Eq(F_95(x), F_107(x) + 2*F_21(x) + F_96(x) + F_99(x))
Eq(F_96(x), F_12(x)*F_97(x))
Eq(F_97(x), F_79(x) + F_98(x))
Eq(F_98(x), F_96(x))
Eq(F_99(x), F_100(x)*F_12(x))
Eq(F_100(x), F_101(x) + F_104(x))
Eq(F_101(x), F_102(x) + F_74(x))
Eq(F_102(x), F_103(x))
Eq(F_103(x), F_101(x)*F_12(x))
Eq(F_104(x), F_105(x) + F_95(x))
Eq(F_105(x), F_106(x))
Eq(F_106(x), F_104(x)*F_12(x))
Eq(F_107(x), F_12(x)*F_82(x))
Eq(F_108(x), F_109(x)*F_12(x))
Eq(F_109(x), F_110(x) + F_78(x))
Eq(F_110(x), F_111(x) + F_119(x))
Eq(F_111(x), F_112(x) + F_114(x) + F_21(x))
Eq(F_112(x), F_113(x)*F_12(x))
Eq(F_113(x), F_24(x) + F_85(x))
Eq(F_114(x), F_115(x)*F_12(x))
Eq(F_115(x), F_116(x) + F_17(x))
Eq(F_116(x), F_111(x) + F_117(x))
Eq(F_117(x), F_118(x))
Eq(F_118(x), F_116(x)*F_12(x))
Eq(F_119(x), F_120(x) + F_130(x) + F_135(x) + F_21(x))
Eq(F_120(x), F_12(x)*F_121(x))
Eq(F_121(x), F_122(x) + F_126(x))
Eq(F_122(x), F_123(x) + F_40(x))
Eq(F_123(x), F_124(x) + 2*F_21(x) + F_53(x))
Eq(F_124(x), F_12(x)*F_125(x))
Eq(F_125(x), F_123(x) + F_25(x))
Eq(F_126(x), F_127(x) + F_74(x))
Eq(F_127(x), F_128(x) + 3*F_21(x) + F_96(x))
Eq(F_128(x), F_12(x)*F_129(x))
Eq(F_129(x), F_127(x) + F_86(x))
Eq(F_130(x), F_12(x)*F_131(x))
Eq(F_131(x), F_132(x) + F_68(x))
Eq(F_132(x), F_119(x) + F_133(x))
Eq(F_133(x), F_134(x))
Eq(F_134(x), F_12(x)*F_132(x))
Eq(F_135(x), F_110(x)*F_12(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": [1, 3, 2, 0], "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": [1, 3, 2, 0], "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": 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