0132_0231_1032_2130
Counting sequence:
1, 1, 2, 6, 20, 66, 214, 686, 2184, 6924, 21894, 69116, 217962, 686906, 2163878, 6814804, 21458584, 67561994, 212703412, 669619018, 2107993366, 6635950894, 20889704640, 65759483532, 207005819678, 651636629908, 2051292738434, 6457275272298, 20326876281014, 63987005600028, 201424735938232, 634064937451586, 1995972816027132, 6283121724562914, 19778634461363670, 62261147635276718, 195991813176546072, 616962452851652396, 1942135536692117270, 6113646635749796332, 19245142454493962650, 60581765584857532346, 190705281886575944710, 600320973795980958340, 1889749816208565159352, 5948741627531916962650, 18726038039695999841988, 58947677101192906661562, 185561335930807730127350, 584128350494532572455534, 1838777071357268750279312, 5788284570017022502233324, 18220935416772291307261550, 57357665028958316426046852, 180556138436659314191478706, 568372494078786398608721066, 1789179226038629567913865494, 5632155560363825589275518476, 17729457057436214450478680472, 55810540774728039399020995922, 175685947487039397427961388684, 553041624681553801153337103314, 1740919197040113282104636954870, 5480237861603407583138801321710, 17251235479975997208899238372520, 54305147532136185211294400173772, 170947121550224768487637114287142, 538124279084475738897601349775580, 1693960899219383282082862165966282, 5332417881174367573614429147928762, 16785913106123869974083026281798246, 52840359680191908031524269525785908, 166336117283570684925367607758503512, 523609303199831475876791007444106282, 1648269220628806937354206193438003924, 5188585090199306367895060794000807210, 16333142002110640857760898472778537046, 51415081958473460709445583441840157870, 161849486917761245521860630621606682336, 509485843797735278595798533590846502348, 1603809996396130914084921551854729539390, 5048631941109079217189878689096632171124, 15892583619045659707332003538070480245218, 50028248649251532696306209444161341359978, 157483875680978232089344525466159731581558, 495743340395203992393643116299483731903868, 1560549983187132810838548376702133773055480, 4912453787243897507934015240293277025741282, 15463908539809395610956872061200257311001372, 48678822780693022769549617128291465530434754, 153236019290587801830633120025194634395671446, 482371517360081534779583005687776932664773422, 1518456834349255756705467289625160361727295160, 4779948804607366224932723281105195420760093932, 15046796232741779165632014300518457805422387894, 47365795351494273746911234726954165870934813388, 149102741512359123720633379444508006587828680698, 469360376227695452817335348211971028743125664314, 1477499075725198713874159529869880345478337416134, 4651017915730067620804879535545248471068573327460, 14640934811972369794493603063297588214175950676088
Generating function in Maple syntax:
(x^4-2*x^3+7*x^2-5*x+1)/(2*x-1)/(x^3-3*x^2+4*x-1)
Generating function in latex syntax:
\frac{x^{4}-2 x^{3}+7 x^{2}-5 x +1}{\left(2 x -1\right) \left(x^{3}-3 x^{2}+4 x -1\right)}
Generating function in sympy syntax:
(x**4 - 2*x**3 + 7*x**2 - 5*x + 1)/((2*x - 1)*(x**3 - 3*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^3-3*x^2+4*x-1)*F(x)-x^4+2*x^3-7*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{3}-3 x^{2}+4 x -1\right) F \! \left(x \right)-x^{4}+2 x^{3}-7 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/1674*(15*2^(2/3)*((I*3^(1/3)-1/3*3^(5/6))*31^(1/2)-31/5*I*3^(5/6)+31/5*3^(1/3))*(9+31^(1/2)*3^(1/2))^(1/3)+372-24*2^(1/3)*((I*3^(2/3)+3^(1/6))*31^(1/2)-31/4*I*3^(1/6)-31/12*3^(2/3))*(9+31^(1/2)*3^(1/2))^(2/3))*(1/12*(108+12*31^(1/2)*3^(1/2))^(1/3)-1/12*I*(36+4*31^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1+1/24*((-I*31^(1/2)+3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)-2^(1/3)*3^(1/6)*31^(1/2))*(9+31^(1/2)*3^(1/2))^(2/3))^(-n)+1/1674*(-15*((I*3^(1/3)+1/3*3^(5/6))*31^(1/2)-31/5*I*3^(5/6)-31/5*3^(1/3))*2^(2/3)*(9+31^(1/2)*3^(1/2))^(1/3)+372+24*2^(1/3)*((I*3^(2/3)-3^(1/6))*31^(1/2)-31/4*I*3^(1/6)+31/12*3^(2/3))*(9+31^(1/2)*3^(1/2))^(2/3))*(1/12*(108+12*31^(1/2)*3^(1/2))^(1/3)+1/12*I*(36+4*31^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1+1/24*((I*31^(1/2)+3)*18^(1/3)-9*I*2^(1/3)*3^(1/6)-2^(1/3)*3^(1/6)*31^(1/2))*(9+31^(1/2)*3^(1/2))^(2/3))^(-n)+1/1674*(10*2^(2/3)*(31^(1/2)*3^(5/6)-93/5*3^(1/3))*(9+31^(1/2)*3^(1/2))^(1/3)+372+48*2^(1/3)*(3^(1/6)*31^(1/2)-31/12*3^(2/3))*(9+31^(1/2)*3^(1/2))^(2/3))*(-1/6*(108+12*31^(1/2)*3^(1/2))^(1/3)+1+1/12*(2^(1/3)*3^(1/6)*31^(1/2)-3*18^(1/3))*(9+31^(1/2)*3^(1/2))^(2/3))^(-n)-1/6*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(15 \,2^{\frac{2}{3}} \left(\left(\mathrm{I} \,3^{\frac{1}{3}}-\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{31}-\frac{31 \,\mathrm{I} \,3^{\frac{5}{6}}}{5}+\frac{31 \,3^{\frac{1}{3}}}{5}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+372-24 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{31}-\frac{31 \,\mathrm{I} \,3^{\frac{1}{6}}}{4}-\frac{31 \,3^{\frac{2}{3}}}{12}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(108+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+1+\frac{\left(\left(-\mathrm{I} \sqrt{31}+3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}-2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{31}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}\right)^{-n}}{1674}\\+\\\frac{\left(-15 \left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{31}-\frac{31 \,\mathrm{I} \,3^{\frac{5}{6}}}{5}-\frac{31 \,3^{\frac{1}{3}}}{5}\right) 2^{\frac{2}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+372+24 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{31}-\frac{31 \,\mathrm{I} \,3^{\frac{1}{6}}}{4}+\frac{31 \,3^{\frac{2}{3}}}{12}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(108+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \left(36+4 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+1+\frac{\left(\left(\mathrm{I} \sqrt{31}+3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}-2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{31}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}\right)^{-n}}{1674}\\+\\\frac{\left(10 \,2^{\frac{2}{3}} \left(\sqrt{31}\, 3^{\frac{5}{6}}-\frac{93 \,3^{\frac{1}{3}}}{5}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+372+48 \,2^{\frac{1}{3}} \left(3^{\frac{1}{6}} \sqrt{31}-\frac{31 \,3^{\frac{2}{3}}}{12}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{\left(108+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+1+\frac{\left(2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{31}-3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}\right)^{-n}}{1674}\\-\frac{2^{n}}{6} & \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(n+4) = -2*a(n)+7*a(n+1)-11*a(n+2)+6*a(n+3), n >= 5
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(n +4\right) = -2 a \! \left(n \right)+7 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15335/
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[77,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[49,x]+F[59,x]
F[49,x] = F[10,x]+F[50,x]
F[50,x] = F[18,x]+F[21,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[17,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[12,x]*F[56,x]
F[59,x] = F[60,x]+F[63,x]
F[60,x] = F[18,x]+F[47,x]+F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[23,x]+F[60,x]
F[63,x] = 2*F[18,x]+F[64,x]+F[73,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[66,x]+F[69,x]
F[66,x] = F[60,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[12,x]*F[66,x]
F[69,x] = F[70,x]+F[75,x]
F[70,x] = 2*F[18,x]+F[71,x]+F[73,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[60,x]+F[70,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[41,x]+F[70,x]
F[75,x] = F[76,x]
F[76,x] = F[12,x]*F[69,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[62,x]+F[79,x]
F[79,x] = F[111,x]+F[80,x]
F[80,x] = F[109,x]+F[18,x]+F[81,x]
F[81,x] = F[12,x]*F[82,x]
F[82,x] = F[83,x]+F[95,x]
F[83,x] = F[12,x]+F[84,x]
F[84,x] = F[18,x]+F[85,x]+F[92,x]
F[85,x] = F[12,x]*F[86,x]
F[86,x] = F[87,x]+F[90,x]
F[87,x] = F[12,x]+F[88,x]
F[88,x] = F[89,x]
F[89,x] = F[12,x]*F[87,x]
F[90,x] = F[91,x]+F[93,x]
F[91,x] = F[92,x]
F[92,x] = F[12,x]*F[14,x]
F[93,x] = F[94,x]
F[94,x] = F[12,x]*F[90,x]
F[95,x] = F[96,x]+F[98,x]
F[96,x] = F[97,x]
F[97,x] = F[12,x]*F[23,x]
F[98,x] = 2*F[18,x]+F[106,x]+F[99,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[96,x]
F[102,x] = F[103,x]
F[103,x] = F[101,x]*F[12,x]
F[104,x] = F[105,x]+F[107,x]
F[105,x] = F[106,x]
F[106,x] = F[12,x]*F[41,x]
F[107,x] = F[108,x]
F[108,x] = F[104,x]*F[12,x]
F[109,x] = F[110,x]*F[12,x]
F[110,x] = F[23,x]+F[80,x]
F[111,x] = F[112,x]+F[142,x]+F[144,x]+F[18,x]
F[112,x] = F[113,x]*F[12,x]
F[113,x] = F[114,x]+F[128,x]
F[114,x] = F[115,x]+F[117,x]
F[115,x] = F[116,x]
F[116,x] = F[10,x]*F[12,x]
F[117,x] = 2*F[18,x]+F[118,x]+F[125,x]
F[118,x] = F[119,x]*F[12,x]
F[119,x] = F[120,x]+F[123,x]
F[120,x] = F[115,x]+F[121,x]
F[121,x] = F[122,x]
F[122,x] = F[12,x]*F[120,x]
F[123,x] = F[124,x]+F[126,x]
F[124,x] = F[125,x]
F[125,x] = F[12,x]*F[17,x]
F[126,x] = F[127,x]
F[127,x] = F[12,x]*F[123,x]
F[128,x] = F[129,x]+F[131,x]
F[129,x] = F[130,x]
F[130,x] = F[12,x]*F[60,x]
F[131,x] = 3*F[18,x]+F[132,x]+F[139,x]
F[132,x] = F[12,x]*F[133,x]
F[133,x] = F[134,x]+F[137,x]
F[134,x] = F[129,x]+F[135,x]
F[135,x] = F[136,x]
F[136,x] = F[12,x]*F[134,x]
F[137,x] = F[138,x]+F[140,x]
F[138,x] = F[139,x]
F[139,x] = F[12,x]*F[70,x]
F[140,x] = F[141,x]
F[141,x] = F[12,x]*F[137,x]
F[142,x] = F[12,x]*F[143,x]
F[143,x] = F[111,x]+F[60,x]
F[144,x] = F[12,x]*F[79,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_{77}\! \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_{59}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \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_{17}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{60}\! \left(x \right) = F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{23}\! \left(x \right)+F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{73}\! \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_{69}\! \left(x \right)
F_{66}\! \left(x \right) = F_{60}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{75}\! \left(x \right)
F_{70}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{60}\! \left(x \right)+F_{70}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{41}\! \left(x \right)+F_{70}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{62}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{111}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{109}\! \left(x \right)+F_{18}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \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) = F_{12}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{18}\! \left(x \right)+F_{85}\! \left(x \right)+F_{92}\! \left(x \right)
F_{85}\! \left(x \right) = F_{12}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)+F_{90}\! \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_{89}\! \left(x \right) = F_{12}\! \left(x \right) F_{87}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)+F_{93}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{12}\! \left(x \right) F_{90}\! \left(x \right)
F_{95}\! \left(x \right) = F_{96}\! \left(x \right)+F_{98}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{98}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{106}\! \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_{104}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{96}\! \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_{107}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)
F_{106}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)
F_{108}\! \left(x \right) = F_{104}\! \left(x \right) F_{12}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right) F_{12}\! \left(x \right)
F_{110}\! \left(x \right) = F_{23}\! \left(x \right)+F_{80}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)+F_{142}\! \left(x \right)+F_{144}\! \left(x \right)+F_{18}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{12}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)+F_{128}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{117}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)
F_{116}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{117}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{118}\! \left(x \right)+F_{125}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right) F_{12}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)+F_{123}\! \left(x \right)
F_{120}\! \left(x \right) = F_{115}\! \left(x \right)+F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)
F_{122}\! \left(x \right) = F_{12}\! \left(x \right) F_{120}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)+F_{126}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)
F_{125}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)
F_{127}\! \left(x \right) = F_{12}\! \left(x \right) F_{123}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right)+F_{131}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)
F_{130}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{131}\! \left(x \right) = 3 F_{18}\! \left(x \right)+F_{132}\! \left(x \right)+F_{139}\! \left(x \right)
F_{132}\! \left(x \right) = F_{12}\! \left(x \right) F_{133}\! \left(x \right)
F_{133}\! \left(x \right) = F_{134}\! \left(x \right)+F_{137}\! \left(x \right)
F_{134}\! \left(x \right) = F_{129}\! \left(x \right)+F_{135}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right)
F_{136}\! \left(x \right) = F_{12}\! \left(x \right) F_{134}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right)+F_{140}\! \left(x \right)
F_{138}\! \left(x \right) = F_{139}\! \left(x \right)
F_{139}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right)
F_{141}\! \left(x \right) = F_{12}\! \left(x \right) F_{137}\! \left(x \right)
F_{142}\! \left(x \right) = F_{12}\! \left(x \right) F_{143}\! \left(x \right)
F_{143}\! \left(x \right) = F_{111}\! \left(x \right)+F_{60}\! \left(x \right)
F_{144}\! \left(x \right) = F_{12}\! \left(x \right) F_{79}\! \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_77(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_59(x))
Eq(F_49(x), F_10(x) + F_50(x))
Eq(F_50(x), F_18(x) + F_21(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_17(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_12(x)*F_56(x))
Eq(F_59(x), F_60(x) + F_63(x))
Eq(F_60(x), F_18(x) + F_47(x) + F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_23(x) + F_60(x))
Eq(F_63(x), 2*F_18(x) + F_64(x) + F_73(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_69(x))
Eq(F_66(x), F_60(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_12(x)*F_66(x))
Eq(F_69(x), F_70(x) + F_75(x))
Eq(F_70(x), 2*F_18(x) + F_71(x) + F_73(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_60(x) + F_70(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_41(x) + F_70(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_12(x)*F_69(x))
Eq(F_77(x), F_12(x)*F_78(x))
Eq(F_78(x), F_62(x) + F_79(x))
Eq(F_79(x), F_111(x) + F_80(x))
Eq(F_80(x), F_109(x) + F_18(x) + F_81(x))
Eq(F_81(x), F_12(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_95(x))
Eq(F_83(x), F_12(x) + F_84(x))
Eq(F_84(x), F_18(x) + F_85(x) + F_92(x))
Eq(F_85(x), F_12(x)*F_86(x))
Eq(F_86(x), F_87(x) + F_90(x))
Eq(F_87(x), F_12(x) + F_88(x))
Eq(F_88(x), F_89(x))
Eq(F_89(x), F_12(x)*F_87(x))
Eq(F_90(x), F_91(x) + F_93(x))
Eq(F_91(x), F_92(x))
Eq(F_92(x), F_12(x)*F_14(x))
Eq(F_93(x), F_94(x))
Eq(F_94(x), F_12(x)*F_90(x))
Eq(F_95(x), F_96(x) + F_98(x))
Eq(F_96(x), F_97(x))
Eq(F_97(x), F_12(x)*F_23(x))
Eq(F_98(x), F_106(x) + 2*F_18(x) + F_99(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_96(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_107(x))
Eq(F_105(x), F_106(x))
Eq(F_106(x), F_12(x)*F_41(x))
Eq(F_107(x), F_108(x))
Eq(F_108(x), F_104(x)*F_12(x))
Eq(F_109(x), F_110(x)*F_12(x))
Eq(F_110(x), F_23(x) + F_80(x))
Eq(F_111(x), F_112(x) + F_142(x) + F_144(x) + F_18(x))
Eq(F_112(x), F_113(x)*F_12(x))
Eq(F_113(x), F_114(x) + F_128(x))
Eq(F_114(x), F_115(x) + F_117(x))
Eq(F_115(x), F_116(x))
Eq(F_116(x), F_10(x)*F_12(x))
Eq(F_117(x), F_118(x) + F_125(x) + 2*F_18(x))
Eq(F_118(x), F_119(x)*F_12(x))
Eq(F_119(x), F_120(x) + F_123(x))
Eq(F_120(x), F_115(x) + F_121(x))
Eq(F_121(x), F_122(x))
Eq(F_122(x), F_12(x)*F_120(x))
Eq(F_123(x), F_124(x) + F_126(x))
Eq(F_124(x), F_125(x))
Eq(F_125(x), F_12(x)*F_17(x))
Eq(F_126(x), F_127(x))
Eq(F_127(x), F_12(x)*F_123(x))
Eq(F_128(x), F_129(x) + F_131(x))
Eq(F_129(x), F_130(x))
Eq(F_130(x), F_12(x)*F_60(x))
Eq(F_131(x), F_132(x) + F_139(x) + 3*F_18(x))
Eq(F_132(x), F_12(x)*F_133(x))
Eq(F_133(x), F_134(x) + F_137(x))
Eq(F_134(x), F_129(x) + F_135(x))
Eq(F_135(x), F_136(x))
Eq(F_136(x), F_12(x)*F_134(x))
Eq(F_137(x), F_138(x) + F_140(x))
Eq(F_138(x), F_139(x))
Eq(F_139(x), F_12(x)*F_70(x))
Eq(F_140(x), F_141(x))
Eq(F_141(x), F_12(x)*F_137(x))
Eq(F_142(x), F_12(x)*F_143(x))
Eq(F_143(x), F_111(x) + F_60(x))
Eq(F_144(x), F_12(x)*F_79(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, 3, 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, 0, 3, 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": 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