0123_0231_1032_3021_3102
Counting sequence:
1, 1, 2, 6, 19, 49, 111, 243, 512, 1047, 2100, 4146, 8086, 15620, 29941, 57032, 108075, 203923, 383395, 718641, 1343580, 2506493, 4667220, 8676684, 16108302, 29869510, 55329713, 102400054, 189367031, 349957089, 646352227, 1193164699, 2201590428, 4060712127, 7487188448, 13800816978, 25431764730, 46854143316, 86304145369, 158941846928, 292669349479, 538836349163, 991937766831, 1825854694129, 3360530260048, 6184635399597, 11381234482296, 20942926949060, 38535537766602, 70902966940734, 130451440334829, 240003221463382, 441540913838603, 812292137157697, 1494316119080231, 2748925578217971, 5056790089217896, 9302064449419543, 17111069034520860, 31475245153727202, 57896989135901966, 106497235402953300, 195892012269620109, 360322711464316296, 662770976369768275, 1219081191992424179, 2242329388948106827, 4124431558320616865, 7586246649393063540, 13953662107804020493, 25665399336791849836, 47207021626405316076, 86828855624691717558, 159705762673995796838, 293748898564890275001, 540295261589482147302, 993768926194070021583, 1827843834440048604241, 3361957773680908735931, 6183651033863941484923, 11373582892991120911076, 20919402451091107341087, 38476977379507528032232, 70770514475706250788706, 130167787059982738961490, 239416723420990865086212, 440357362216152054939665, 809945654462392206394416, 1489725859124273873930607, 2740038776592823430182059, 5039726309994233552934775, 9269516866316080194392513, 17049323893322630557282120, 31358634930657187021727261, 57677585491739633870292528, 106085721978187430263310404, 195122229864495966066229330, 358886002460802723924739902, 660094706893777528890086421, 1214104156935747321241770086, 2233086836597290085053117635
Generating function in Maple syntax:
-(3*x^8-5*x^6+2*x^5+6*x^4+3*x^3-2*x+1)/(x-1)/(x^2+x-1)/(x^3+x^2+x-1)
Generating function in latex syntax:
-\frac{3 x^{8}-5 x^{6}+2 x^{5}+6 x^{4}+3 x^{3}-2 x +1}{\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(-3*x**8 + 5*x**6 - 2*x**5 - 6*x**4 - 3*x**3 + 2*x - 1)/((x - 1)*(x**2 + x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2+x-1)*(x^3+x^2+x-1)*F(x)+3*x^8-5*x^6+2*x^5+6*x^4+3*x^3-2*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+3 x^{8}-5 x^{6}+2 x^{5}+6 x^{4}+3 x^{3}-2 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,1/2640*(((3740*I-760*11^(1/2))*3^(1/2)-2280*I*11^(1/2)+3740)*(17+3*11^(1/2)*3^(1/2))^(1/3)+6160+((2915*I+425*11^(1/2))*3^(1/2)-1275*I*11^(1/2)-2915)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(1/24*((17*I+3*11^(1/2))*3^(1/2)-9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/2640*(((-2915*I+425*11^(1/2))*3^(1/2)+1275*I*11^(1/2)-2915)*(17+3*11^(1/2)*3^(1/2))^(2/3)+6160+((-3740*I-760*11^(1/2))*3^(1/2)+2280*I*11^(1/2)+3740)*(17+3*11^(1/2)*3^(1/2))^(1/3))*(1/24*((-17*I+3*11^(1/2))*3^(1/2)+9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/2640*((-850*11^(1/2)*3^(1/2)+5830)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1520*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-7480*(17+3*11^(1/2)*3^(1/2))^(1/3)+6160)*(1/3*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3+17/12*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/4*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2))^(-n)+1/2640*(7128*5^(1/2)-11880)*(-1/2*5^(1/2)-1/2)^(-n)+4+1/2640*(-7128*5^(1/2)-11880)*(1/2*5^(1/2)-1/2)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(\left(\left(3740 \,\mathrm{I}-760 \sqrt{11}\right) \sqrt{3}-2280 \,\mathrm{I} \sqrt{11}+3740\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+6160+\left(\left(2915 \,\mathrm{I}+425 \sqrt{11}\right) \sqrt{3}-1275 \,\mathrm{I} \sqrt{11}-2915\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{2640}\\+\\\frac{\left(\left(\left(-2915 \,\mathrm{I}+425 \sqrt{11}\right) \sqrt{3}+1275 \,\mathrm{I} \sqrt{11}-2915\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+6160+\left(\left(-3740 \,\mathrm{I}-760 \sqrt{11}\right) \sqrt{3}+2280 \,\mathrm{I} \sqrt{11}+3740\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{2640}\\+\\\frac{\left(\left(-850 \sqrt{11}\, \sqrt{3}+5830\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+1520 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-7480 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+6160\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{2640}\\+\frac{\left(7128 \sqrt{5}-11880\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{2640}+4+\\\frac{\left(-7128 \sqrt{5}-11880\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{2640} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 49
a(6) = 111
a(7) = 243
a(8) = 512
a(n+5) = -a(n)-2*a(n+1)-a(n+2)+a(n+3)+2*a(n+4)+8, n >= 9
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) = 19
a \! \left(5\right) = 49
a \! \left(6\right) = 111
a \! \left(7\right) = 243
a \! \left(8\right) = 512
a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)+8, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17302/
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[24,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[19,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[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[20,x]
F[24,x] = F[25,x]+F[37,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[1,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[17,x]*F[4,x]
F[31,x] = F[25,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[25,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[25,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]+F[74,x]
F[38,x] = 0
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[51,x]
F[41,x] = F[29,x]+F[42,x]
F[42,x] = F[38,x]+F[43,x]+F[47,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[4,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = x^2
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[29,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[22,x]*F[4,x]
F[51,x] = F[52,x]+F[68,x]
F[52,x] = F[38,x]+F[39,x]+F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[25,x]+F[55,x]
F[55,x] = F[36,x]+F[38,x]+F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = F[4,x]+F[59,x]
F[59,x] = F[38,x]+F[43,x]+F[60,x]
F[60,x] = F[29,x]*F[4,x]
F[61,x] = F[55,x]+F[62,x]
F[62,x] = 2*F[38,x]+F[63,x]+F[67,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[55,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[35,x]*F[4,x]
F[67,x] = F[32,x]*F[4,x]
F[68,x] = 2*F[38,x]+F[63,x]+F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[32,x]+F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[55,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[76,x]+F[88,x]
F[76,x] = F[25,x]+F[77,x]
F[77,x] = F[38,x]+F[56,x]+F[78,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[25,x]+F[80,x]
F[80,x] = F[38,x]+F[78,x]+F[81,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[83,x]+F[84,x]
F[83,x] = F[4,x]+F[49,x]
F[84,x] = F[35,x]+F[85,x]
F[85,x] = F[86,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[35,x]
F[88,x] = F[89,x]+F[99,x]
F[89,x] = F[38,x]+F[90,x]+F[96,x]
F[90,x] = F[4,x]*F[91,x]
F[91,x] = F[92,x]+F[94,x]
F[92,x] = F[29,x]+F[93,x]
F[93,x] = F[50,x]
F[94,x] = F[32,x]+F[95,x]
F[95,x] = F[86,x]
F[96,x] = F[4,x]*F[97,x]
F[97,x] = F[34,x]+F[98,x]
F[98,x] = F[89,x]
F[99,x] = 2*F[38,x]+F[100,x]+F[104,x]
F[100,x] = F[101,x]*F[4,x]
F[101,x] = F[102,x]+F[103,x]
F[102,x] = F[49,x]
F[103,x] = F[71,x]
F[104,x] = F[105,x]*F[4,x]
F[105,x] = F[106,x]+F[73,x]
F[106,x] = F[107,x]
F[107,x] = 2*F[38,x]+F[100,x]+F[104,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{24}\! \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_{19}\! \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_{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_{20}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{37}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{25}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{74}\! \left(x \right)
F_{38}\! \left(x \right) = 0
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{51}\! \left(x \right)
F_{41}\! \left(x \right) = F_{29}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = x^{2}
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{29}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{68}\! \left(x \right)
F_{52}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \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_{36}\! \left(x \right)+F_{38}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \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_{4}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{61}\! \left(x \right) = F_{55}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = 2 F_{38}\! \left(x \right)+F_{63}\! \left(x \right)+F_{67}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{55}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{67}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{68}\! \left(x \right) = 2 F_{38}\! \left(x \right)+F_{63}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{32}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{55}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{88}\! \left(x \right)
F_{76}\! \left(x \right) = F_{25}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{38}\! \left(x \right)+F_{56}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{25}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{38}\! \left(x \right)+F_{78}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{84}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right)+F_{49}\! \left(x \right)
F_{84}\! \left(x \right) = F_{35}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{35}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{99}\! \left(x \right)
F_{89}\! \left(x \right) = F_{38}\! \left(x \right)+F_{90}\! \left(x \right)+F_{96}\! \left(x \right)
F_{90}\! \left(x \right) = F_{4}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)+F_{94}\! \left(x \right)
F_{92}\! \left(x \right) = F_{29}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{50}\! \left(x \right)
F_{94}\! \left(x \right) = F_{32}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{86}\! \left(x \right)
F_{96}\! \left(x \right) = F_{4}\! \left(x \right) F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{34}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{89}\! \left(x \right)
F_{99}\! \left(x \right) = 2 F_{38}\! \left(x \right)+F_{100}\! \left(x \right)+F_{104}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{4}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{103}\! \left(x \right)
F_{102}\! \left(x \right) = F_{49}\! \left(x \right)
F_{103}\! \left(x \right) = F_{71}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right) F_{4}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{73}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)
F_{107}\! \left(x \right) = 2 F_{38}\! \left(x \right)+F_{100}\! \left(x \right)+F_{104}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_24(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_19(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))
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_20(x))
Eq(F_24(x), F_25(x) + F_37(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_1(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_17(x)*F_4(x))
Eq(F_31(x), F_25(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_25(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_25(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_39(x) + F_74(x))
Eq(F_38(x), 0)
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_51(x))
Eq(F_41(x), F_29(x) + F_42(x))
Eq(F_42(x), F_38(x) + F_43(x) + F_47(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_4(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), x**2)
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_29(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_22(x)*F_4(x))
Eq(F_51(x), F_52(x) + F_68(x))
Eq(F_52(x), F_38(x) + F_39(x) + F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_25(x) + F_55(x))
Eq(F_55(x), F_36(x) + F_38(x) + F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), F_4(x) + F_59(x))
Eq(F_59(x), F_38(x) + F_43(x) + F_60(x))
Eq(F_60(x), F_29(x)*F_4(x))
Eq(F_61(x), F_55(x) + F_62(x))
Eq(F_62(x), 2*F_38(x) + F_63(x) + F_67(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_55(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_35(x)*F_4(x))
Eq(F_67(x), F_32(x)*F_4(x))
Eq(F_68(x), 2*F_38(x) + F_63(x) + F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_32(x) + F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_55(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_88(x))
Eq(F_76(x), F_25(x) + F_77(x))
Eq(F_77(x), F_38(x) + F_56(x) + F_78(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_25(x) + F_80(x))
Eq(F_80(x), F_38(x) + F_78(x) + F_81(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_84(x))
Eq(F_83(x), F_4(x) + F_49(x))
Eq(F_84(x), F_35(x) + F_85(x))
Eq(F_85(x), F_86(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_35(x))
Eq(F_88(x), F_89(x) + F_99(x))
Eq(F_89(x), F_38(x) + F_90(x) + F_96(x))
Eq(F_90(x), F_4(x)*F_91(x))
Eq(F_91(x), F_92(x) + F_94(x))
Eq(F_92(x), F_29(x) + F_93(x))
Eq(F_93(x), F_50(x))
Eq(F_94(x), F_32(x) + F_95(x))
Eq(F_95(x), F_86(x))
Eq(F_96(x), F_4(x)*F_97(x))
Eq(F_97(x), F_34(x) + F_98(x))
Eq(F_98(x), F_89(x))
Eq(F_99(x), F_100(x) + F_104(x) + 2*F_38(x))
Eq(F_100(x), F_101(x)*F_4(x))
Eq(F_101(x), F_102(x) + F_103(x))
Eq(F_102(x), F_49(x))
Eq(F_103(x), F_71(x))
Eq(F_104(x), F_105(x)*F_4(x))
Eq(F_105(x), F_106(x) + F_73(x))
Eq(F_106(x), F_107(x))
Eq(F_107(x), F_100(x) + F_104(x) + 2*F_38(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, 2, 3], "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": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "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": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "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": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 1], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [3, 0, 2, 1], "pos": [[1, 2], [1, 0], [1, 2], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 0], [1, 0], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 1], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 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