0132_0231_3012_3021

Counting sequence:
1, 1, 2, 6, 20, 65, 201, 598, 1738, 4989, 14235, 40504, 115108, 326951, 928457, 2636330, 7485498, 21253705, 60345623, 171338820, 486480368, 1381257627, 3921786885, 11135077998, 31615679958, 89765981885, 254871364003, 723652888712, 2054658062956, 5833763425359, 16563727227441, 47029171334674, 133529303280114, 379127982237137, 1076453058497967, 3056358911604108, 8677879376899432, 24638987978177859, 69957152228322589, 198628415754378486, 563963030060580046, 1601252761681452197, 4546415757992246395, 12908578045364337744, 36651154672851270692, 104063138025932225527, 295465089503052495817, 838910115254183623578, 2381906379056388963178, 6762915234214926351161, 19201855651142516800263, 54519574425823139723988, 154796705567152344398496, 439512235867601852299883, 1247901270053444311054901, 3543149547877674523818494, 10060017583032124555655622, 28563274680724688749459853, 81099327486525605822855507, 230264246389267917404895512, 653786224972418623827081436, 1856286569305654037334456479, 5270529870111477488839293025, 14964545653168131053673510946, 42488636270837922334005786594, 120637422210902120099007690657, 342524235066593053167354640223, 972524523964439576941399373596, 2761276000013190335294194317016, 7840054374328189139745889056403, 22260162545188874371689577707085, 63202984683469121598363143239174, 179451397301777367186186040243646, 509513975563610731094670743434613, 1446656282414271698830665099244843, 4107472022006198605316349503730912, 11662290909494929088132079104816276, 33112588114783465328924248602833991, 94016132865166542173801376999535609, 266938760820522165555772671796442026, 757915688050989424185733676393296730, 2151939974652202118867446772386493193, 6109974668045895109838397727565673143, 17347970149676750812255271173503316580, 49255861875827827047402387328523563536, 139851516240696095160265260075212337595, 397078557758826574075962033333761732965, 1127419889824177569757591867691244712206, 3201068360742796173047049857580356595894, 9088750999192127561309756026223971004509, 25805570333445680692566008409956947062787, 73269414058499804871119109250092617428904, 208032876898600751836831915412990623477580, 590664992027295149247983241471505918324463, 1677067288631777199386843522734246414249617, 4761674940215129704229104955876943230660146, 13519760590388000317865012215513799102295506, 38386477177955036853317316084015916315045233, 108990216230698405550267425385762355312323855, 309454477391749475423530461638013354193710252, 878629999000114353199686777177035828155640904

Generating function in Maple syntax:
-(x^2-x+1)*(3*x^3-8*x^2+5*x-1)/(2*x^3-4*x^2+4*x-1)/(x-1)^3

Generating function in latex syntax:
-\frac{\left(x^{2}-x +1\right) \left(3 x^{3}-8 x^{2}+5 x -1\right)}{\left(2 x^{3}-4 x^{2}+4 x -1\right) \left(x -1\right)^{3}}

Generating function in sympy syntax:
(-x**2 + x - 1)*(3*x**3 - 8*x**2 + 5*x - 1)/((x - 1)**3*(2*x**3 - 4*x**2 + 4*x - 1))

Implicit equation for the generating function in Maple syntax:
(2*x^3-4*x^2+4*x-1)*(x-1)^3*F(x)+(x^2-x+1)*(3*x^3-8*x^2+5*x-1) = 0

Implicit equation for the generating function in latex syntax:
\left(2 x^{3}-4 x^{2}+4 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(x^{2}-x +1\right) \left(3 x^{3}-8 x^{2}+5 x -1\right) = 0

Explicit closed form in Maple syntax:
1/192*(2*2^(1/3)*((3*I-3^(1/2))*11^(1/2)+I*3^(1/2)-1)*(13+3*11^(1/2)*3^(1/2))^(1/3)-64-7*2^(2/3)*((-3/7*I-1/7*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)+1/192*(-2*2^(1/3)*((3*I+3^(1/2))*11^(1/2)+I*3^(1/2)+1)*(13+3*11^(1/2)*3^(1/2))^(1/3)-64+7*((-3/7*I+1/7*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)+1/192*((4*2^(1/3)*3^(1/2)*11^(1/2)+4*2^(1/3))*(13+3*11^(1/2)*3^(1/2))^(1/3)-64+(-2*2^(2/3)*3^(1/2)*11^(1/2)+14*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)-1/2*n^2-1/2*n+2

Explicit closed form in latex syntax:
\frac{\left(2 \,2^{\frac{1}{3}} \left(\left(3 \,\mathrm{I}-\sqrt{3}\right) \sqrt{11}+\mathrm{I} \sqrt{3}-1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-64-7 \,2^{\frac{2}{3}} \left(\left(-\frac{3 \,\mathrm{I}}{7}-\frac{\sqrt{3}}{7}\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}}{192}+\frac{\left(-2 \,2^{\frac{1}{3}} \left(\left(3 \,\mathrm{I}+\sqrt{3}\right) \sqrt{11}+\mathrm{I} \sqrt{3}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-64+7 \left(\left(-\frac{3 \,\mathrm{I}}{7}+\frac{\sqrt{3}}{7}\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}}{192}+\frac{\left(\left(4 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{11}+4 \,2^{\frac{1}{3}}\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-64+\left(-2 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{11}+14 \,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}}{192}-\frac{n^{2}}{2}-\frac{n}{2}+2

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(n+3) = 2*a(n)-4*a(n+1)+4*a(n+2)+1/2*n*(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 +3\right) = 2 a \! \left(n \right)-4 a \! \left(n +1\right)+4 a \! \left(n +2\right)+\frac{n \left(n +3\right)}{2}, \quad n \geq 6

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15406/
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[36,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[24,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[4,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[4,x]
F[23,x] = F[19,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[11,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[11,x]*F[4,x]
F[30,x] = F[24,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[35,x]
F[34,x] = F[28,x]
F[35,x] = F[31,x]
F[36,x] = F[37,x]+F[47,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[17,x]+F[40,x]
F[40,x] = F[37,x]+F[41,x]
F[41,x] = F[42,x]+F[43,x]+F[46,x]
F[42,x] = 0
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[22,x]+F[45,x]
F[45,x] = F[41,x]
F[46,x] = F[37,x]*F[4,x]
F[47,x] = F[137,x]+F[42,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[80,x]
F[50,x] = F[51,x]+F[7,x]
F[51,x] = F[29,x]+F[42,x]+F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[28,x]+F[4,x]
F[55,x] = F[56,x]+F[68,x]
F[56,x] = F[42,x]+F[57,x]+F[58,x]
F[57,x] = x^2
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[62,x]
F[60,x] = F[4,x]+F[61,x]
F[61,x] = F[57,x]
F[62,x] = F[56,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[67,x]
F[66,x] = F[61,x]
F[67,x] = F[63,x]
F[68,x] = 2*F[42,x]+F[69,x]+F[70,x]
F[69,x] = F[28,x]*F[4,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]+F[74,x]
F[72,x] = F[28,x]+F[73,x]
F[73,x] = F[69,x]
F[74,x] = F[68,x]+F[75,x]
F[75,x] = F[76,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[78,x]+F[79,x]
F[78,x] = F[73,x]
F[79,x] = F[75,x]
F[80,x] = F[47,x]+F[81,x]
F[81,x] = F[108,x]+F[42,x]+F[82,x]+F[86,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[84,x]+F[85,x]
F[84,x] = F[51,x]
F[85,x] = F[81,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[106,x]+F[42,x]+F[88,x]
F[88,x] = F[4,x]*F[89,x]
F[89,x] = F[90,x]+F[93,x]
F[90,x] = F[11,x]+F[91,x]
F[91,x] = F[29,x]+F[42,x]+F[92,x]
F[92,x] = F[4,x]*F[54,x]
F[93,x] = F[87,x]+F[94,x]
F[94,x] = F[42,x]+F[86,x]+F[95,x]+F[99,x]
F[95,x] = F[4,x]*F[96,x]
F[96,x] = F[97,x]+F[98,x]
F[97,x] = F[91,x]
F[98,x] = F[94,x]
F[99,x] = F[100,x]*F[4,x]
F[100,x] = F[101,x]+F[41,x]
F[101,x] = 2*F[42,x]+F[102,x]+F[86,x]
F[102,x] = F[103,x]*F[4,x]
F[103,x] = F[104,x]+F[105,x]
F[104,x] = F[28,x]
F[105,x] = F[101,x]
F[106,x] = F[107,x]*F[4,x]
F[107,x] = F[37,x]+F[87,x]
F[108,x] = F[109,x]*F[4,x]
F[109,x] = F[100,x]+F[110,x]
F[110,x] = F[111,x]+F[124,x]
F[111,x] = F[112,x]+F[113,x]+F[114,x]+F[42,x]
F[112,x] = 0
F[113,x] = F[4,x]*F[41,x]
F[114,x] = F[115,x]*F[4,x]
F[115,x] = F[116,x]+F[118,x]
F[116,x] = F[117,x]+F[41,x]
F[117,x] = F[113,x]
F[118,x] = F[111,x]+F[119,x]
F[119,x] = F[120,x]
F[120,x] = F[121,x]*F[4,x]
F[121,x] = F[122,x]+F[123,x]
F[122,x] = F[117,x]
F[123,x] = F[119,x]
F[124,x] = 2*F[42,x]+F[125,x]+F[126,x]+F[127,x]
F[125,x] = 0
F[126,x] = F[101,x]*F[4,x]
F[127,x] = F[128,x]*F[4,x]
F[128,x] = F[129,x]+F[131,x]
F[129,x] = F[101,x]+F[130,x]
F[130,x] = F[126,x]
F[131,x] = F[124,x]+F[132,x]
F[132,x] = F[133,x]
F[133,x] = F[134,x]*F[4,x]
F[134,x] = F[135,x]+F[136,x]
F[135,x] = F[130,x]
F[136,x] = F[132,x]
F[137,x] = F[138,x]*F[4,x]
F[138,x] = F[107,x]+F[139,x]
F[139,x] = F[140,x]+F[148,x]
F[140,x] = F[141,x]+F[42,x]+F[43,x]
F[141,x] = F[142,x]*F[4,x]
F[142,x] = F[143,x]+F[40,x]
F[143,x] = F[140,x]+F[144,x]
F[144,x] = F[145,x]
F[145,x] = F[146,x]*F[4,x]
F[146,x] = F[147,x]+F[45,x]
F[147,x] = F[144,x]
F[148,x] = 2*F[42,x]+F[102,x]+F[149,x]
F[149,x] = F[150,x]*F[4,x]
F[150,x] = F[151,x]+F[152,x]
F[151,x] = F[101,x]+F[87,x]
F[152,x] = F[148,x]+F[153,x]
F[153,x] = F[154,x]
F[154,x] = F[155,x]*F[4,x]
F[155,x] = F[156,x]+F[157,x]
F[156,x] = F[101,x]
F[157,x] = F[153,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_{36}\! \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_{24}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{24}\! \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_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{28}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{47}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{17}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{37}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = 0
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{22}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)
F_{46}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{47}\! \left(x \right) = F_{137}\! \left(x \right)+F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{80}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{7}\! \left(x \right)
F_{51}\! \left(x \right) = F_{29}\! \left(x \right)+F_{42}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{28}\! \left(x \right)+F_{4}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{68}\! \left(x \right)
F_{56}\! \left(x \right) = F_{42}\! \left(x \right)+F_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{57}\! \left(x \right) = x^{2}
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{57}\! \left(x \right)
F_{62}\! \left(x \right) = F_{56}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{61}\! \left(x \right)
F_{67}\! \left(x \right) = F_{63}\! \left(x \right)
F_{68}\! \left(x \right) = 2 F_{42}\! \left(x \right)+F_{69}\! \left(x \right)+F_{70}\! \left(x \right)
F_{69}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{74}\! \left(x \right)
F_{72}\! \left(x \right) = F_{28}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{69}\! \left(x \right)
F_{74}\! \left(x \right) = F_{68}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{79}\! \left(x \right)
F_{78}\! \left(x \right) = F_{73}\! \left(x \right)
F_{79}\! \left(x \right) = F_{75}\! \left(x \right)
F_{80}\! \left(x \right) = F_{47}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{108}\! \left(x \right)+F_{42}\! \left(x \right)+F_{82}\! \left(x \right)+F_{86}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{85}\! \left(x \right)
F_{84}\! \left(x \right) = F_{51}\! \left(x \right)
F_{85}\! \left(x \right) = F_{81}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{106}\! \left(x \right)+F_{42}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{90}\! \left(x \right)+F_{93}\! \left(x \right)
F_{90}\! \left(x \right) = F_{11}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{29}\! \left(x \right)+F_{42}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{93}\! \left(x \right) = F_{87}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{42}\! \left(x \right)+F_{86}\! \left(x \right)+F_{95}\! \left(x \right)+F_{99}\! \left(x \right)
F_{95}\! \left(x \right) = F_{4}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)+F_{98}\! \left(x \right)
F_{97}\! \left(x \right) = F_{91}\! \left(x \right)
F_{98}\! \left(x \right) = F_{94}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right) F_{4}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{41}\! \left(x \right)
F_{101}\! \left(x \right) = 2 F_{42}\! \left(x \right)+F_{102}\! \left(x \right)+F_{86}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right) F_{4}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{105}\! \left(x \right)
F_{104}\! \left(x \right) = F_{28}\! \left(x \right)
F_{105}\! \left(x \right) = F_{101}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right) F_{4}\! \left(x \right)
F_{107}\! \left(x \right) = F_{37}\! \left(x \right)+F_{87}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{4}\! \left(x \right)
F_{109}\! \left(x \right) = F_{100}\! \left(x \right)+F_{110}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{124}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)+F_{113}\! \left(x \right)+F_{114}\! \left(x \right)+F_{42}\! \left(x \right)
F_{112}\! \left(x \right) = 0
F_{113}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right) F_{4}\! \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_{41}\! \left(x \right)
F_{117}\! \left(x \right) = F_{113}\! \left(x \right)
F_{118}\! \left(x \right) = F_{111}\! \left(x \right)+F_{119}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right) F_{4}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)+F_{123}\! \left(x \right)
F_{122}\! \left(x \right) = F_{117}\! \left(x \right)
F_{123}\! \left(x \right) = F_{119}\! \left(x \right)
F_{124}\! \left(x \right) = 2 F_{42}\! \left(x \right)+F_{125}\! \left(x \right)+F_{126}\! \left(x \right)+F_{127}\! \left(x \right)
F_{125}\! \left(x \right) = 0
F_{126}\! \left(x \right) = F_{101}\! \left(x \right) F_{4}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right) F_{4}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right)+F_{131}\! \left(x \right)
F_{129}\! \left(x \right) = F_{101}\! \left(x \right)+F_{130}\! \left(x \right)
F_{130}\! \left(x \right) = F_{126}\! \left(x \right)
F_{131}\! \left(x \right) = F_{124}\! \left(x \right)+F_{132}\! \left(x \right)
F_{132}\! \left(x \right) = F_{133}\! \left(x \right)
F_{133}\! \left(x \right) = F_{134}\! \left(x \right) F_{4}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)+F_{136}\! \left(x \right)
F_{135}\! \left(x \right) = F_{130}\! \left(x \right)
F_{136}\! \left(x \right) = F_{132}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right) F_{4}\! \left(x \right)
F_{138}\! \left(x \right) = F_{107}\! \left(x \right)+F_{139}\! \left(x \right)
F_{139}\! \left(x \right) = F_{140}\! \left(x \right)+F_{148}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right)+F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right) F_{4}\! \left(x \right)
F_{142}\! \left(x \right) = F_{143}\! \left(x \right)+F_{40}\! \left(x \right)
F_{143}\! \left(x \right) = F_{140}\! \left(x \right)+F_{144}\! \left(x \right)
F_{144}\! \left(x \right) = F_{145}\! \left(x \right)
F_{145}\! \left(x \right) = F_{146}\! \left(x \right) F_{4}\! \left(x \right)
F_{146}\! \left(x \right) = F_{147}\! \left(x \right)+F_{45}\! \left(x \right)
F_{147}\! \left(x \right) = F_{144}\! \left(x \right)
F_{148}\! \left(x \right) = 2 F_{42}\! \left(x \right)+F_{102}\! \left(x \right)+F_{149}\! \left(x \right)
F_{149}\! \left(x \right) = F_{150}\! \left(x \right) F_{4}\! \left(x \right)
F_{150}\! \left(x \right) = F_{151}\! \left(x \right)+F_{152}\! \left(x \right)
F_{151}\! \left(x \right) = F_{101}\! \left(x \right)+F_{87}\! \left(x \right)
F_{152}\! \left(x \right) = F_{148}\! \left(x \right)+F_{153}\! \left(x \right)
F_{153}\! \left(x \right) = F_{154}\! \left(x \right)
F_{154}\! \left(x \right) = F_{155}\! \left(x \right) F_{4}\! \left(x \right)
F_{155}\! \left(x \right) = F_{156}\! \left(x \right)+F_{157}\! \left(x \right)
F_{156}\! \left(x \right) = F_{101}\! \left(x \right)
F_{157}\! \left(x \right) = F_{153}\! \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_36(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_24(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_4(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_4(x))
Eq(F_23(x), F_19(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_11(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_11(x)*F_4(x))
Eq(F_30(x), F_24(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_35(x))
Eq(F_34(x), F_28(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_37(x) + F_47(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_17(x) + F_40(x))
Eq(F_40(x), F_37(x) + F_41(x))
Eq(F_41(x), F_42(x) + F_43(x) + F_46(x))
Eq(F_42(x), 0)
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_22(x) + F_45(x))
Eq(F_45(x), F_41(x))
Eq(F_46(x), F_37(x)*F_4(x))
Eq(F_47(x), F_137(x) + F_42(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_80(x))
Eq(F_50(x), F_51(x) + F_7(x))
Eq(F_51(x), F_29(x) + F_42(x) + F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_28(x) + F_4(x))
Eq(F_55(x), F_56(x) + F_68(x))
Eq(F_56(x), F_42(x) + F_57(x) + F_58(x))
Eq(F_57(x), x**2)
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_62(x))
Eq(F_60(x), F_4(x) + F_61(x))
Eq(F_61(x), F_57(x))
Eq(F_62(x), F_56(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_67(x))
Eq(F_66(x), F_61(x))
Eq(F_67(x), F_63(x))
Eq(F_68(x), 2*F_42(x) + F_69(x) + F_70(x))
Eq(F_69(x), F_28(x)*F_4(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_74(x))
Eq(F_72(x), F_28(x) + F_73(x))
Eq(F_73(x), F_69(x))
Eq(F_74(x), F_68(x) + F_75(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_79(x))
Eq(F_78(x), F_73(x))
Eq(F_79(x), F_75(x))
Eq(F_80(x), F_47(x) + F_81(x))
Eq(F_81(x), F_108(x) + F_42(x) + F_82(x) + F_86(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_84(x) + F_85(x))
Eq(F_84(x), F_51(x))
Eq(F_85(x), F_81(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_106(x) + F_42(x) + F_88(x))
Eq(F_88(x), F_4(x)*F_89(x))
Eq(F_89(x), F_90(x) + F_93(x))
Eq(F_90(x), F_11(x) + F_91(x))
Eq(F_91(x), F_29(x) + F_42(x) + F_92(x))
Eq(F_92(x), F_4(x)*F_54(x))
Eq(F_93(x), F_87(x) + F_94(x))
Eq(F_94(x), F_42(x) + F_86(x) + F_95(x) + F_99(x))
Eq(F_95(x), F_4(x)*F_96(x))
Eq(F_96(x), F_97(x) + F_98(x))
Eq(F_97(x), F_91(x))
Eq(F_98(x), F_94(x))
Eq(F_99(x), F_100(x)*F_4(x))
Eq(F_100(x), F_101(x) + F_41(x))
Eq(F_101(x), F_102(x) + 2*F_42(x) + F_86(x))
Eq(F_102(x), F_103(x)*F_4(x))
Eq(F_103(x), F_104(x) + F_105(x))
Eq(F_104(x), F_28(x))
Eq(F_105(x), F_101(x))
Eq(F_106(x), F_107(x)*F_4(x))
Eq(F_107(x), F_37(x) + F_87(x))
Eq(F_108(x), F_109(x)*F_4(x))
Eq(F_109(x), F_100(x) + F_110(x))
Eq(F_110(x), F_111(x) + F_124(x))
Eq(F_111(x), F_112(x) + F_113(x) + F_114(x) + F_42(x))
Eq(F_112(x), 0)
Eq(F_113(x), F_4(x)*F_41(x))
Eq(F_114(x), F_115(x)*F_4(x))
Eq(F_115(x), F_116(x) + F_118(x))
Eq(F_116(x), F_117(x) + F_41(x))
Eq(F_117(x), F_113(x))
Eq(F_118(x), F_111(x) + F_119(x))
Eq(F_119(x), F_120(x))
Eq(F_120(x), F_121(x)*F_4(x))
Eq(F_121(x), F_122(x) + F_123(x))
Eq(F_122(x), F_117(x))
Eq(F_123(x), F_119(x))
Eq(F_124(x), F_125(x) + F_126(x) + F_127(x) + 2*F_42(x))
Eq(F_125(x), 0)
Eq(F_126(x), F_101(x)*F_4(x))
Eq(F_127(x), F_128(x)*F_4(x))
Eq(F_128(x), F_129(x) + F_131(x))
Eq(F_129(x), F_101(x) + F_130(x))
Eq(F_130(x), F_126(x))
Eq(F_131(x), F_124(x) + F_132(x))
Eq(F_132(x), F_133(x))
Eq(F_133(x), F_134(x)*F_4(x))
Eq(F_134(x), F_135(x) + F_136(x))
Eq(F_135(x), F_130(x))
Eq(F_136(x), F_132(x))
Eq(F_137(x), F_138(x)*F_4(x))
Eq(F_138(x), F_107(x) + F_139(x))
Eq(F_139(x), F_140(x) + F_148(x))
Eq(F_140(x), F_141(x) + F_42(x) + F_43(x))
Eq(F_141(x), F_142(x)*F_4(x))
Eq(F_142(x), F_143(x) + F_40(x))
Eq(F_143(x), F_140(x) + F_144(x))
Eq(F_144(x), F_145(x))
Eq(F_145(x), F_146(x)*F_4(x))
Eq(F_146(x), F_147(x) + F_45(x))
Eq(F_147(x), F_144(x))
Eq(F_148(x), F_102(x) + F_149(x) + 2*F_42(x))
Eq(F_149(x), F_150(x)*F_4(x))
Eq(F_150(x), F_151(x) + F_152(x))
Eq(F_151(x), F_101(x) + F_87(x))
Eq(F_152(x), F_148(x) + F_153(x))
Eq(F_153(x), F_154(x))
Eq(F_154(x), F_155(x)*F_4(x))
Eq(F_155(x), F_156(x) + F_157(x))
Eq(F_156(x), F_101(x))
Eq(F_157(x), F_153(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": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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