0132_0321_1023_1302
Counting sequence:
1, 1, 2, 6, 20, 62, 187, 564, 1708, 5179, 15701, 47589, 144233, 437150, 1324957, 4015815, 12171526, 36890632, 111811676, 338889612, 1027139351, 3113153090, 9435644836, 28598462973, 86678981557, 262715022583, 796262044769, 2413387851552, 7314728813465, 22170186022985, 67195539414922, 203662725814926, 617280644630884, 1870913750718118, 5670545955185651, 17186837938162788, 52091527102878860, 157884027630508475, 478530148129474357, 1450375355287477245, 4395937600688253049, 13323631926519155902, 40382549489685009029, 122395328261886174543, 370967573113762905566, 1124364322202392854032, 3407831898703344170620, 10328785804117295148564, 31305480833114859204895, 94883672561188918685650, 287582591901149074685300, 871633073764599152591229, 2641829640166364631736997, 8007112233038583200119727, 24268728512122631887304257, 73556004518698738721855040, 222940637291821942417708081, 675709999221732929013707665, 2048007077554824211236747986, 6207297501214407062979836310, 18813676325076653397709329332, 57022305890043368959224215630, 172828707841843174717222094443, 523825927206088766076862591572, 1587662173950974718650964128428, 4812039740070040050009587317051, 14584794448046327429423078408021, 44205002573122319264966497246293, 133980787967258973444776538841097, 406081902486804664338868024791902, 1230792220505526008750678926633069, 3730403844101738289817116642726231, 11306467987239399771462164980343158, 34268734348586769643726321616317720, 103864987295713415702919787373918108, 314804027373820995832226795687524188, 954138427501328290930052046440994407, 2891894828758516797372324521771609762, 8765033940097343659324963754142634372, 26565910767937399900730020077120687741, 80518526197763282652835024107980512981, 244043319940852750492837179861197819431, 739670046387508897924151322476602313633, 2241863361207754493925166699108537569120, 6794855834533366518234678050142194305481, 20594504826208064258850501527317651192793, 62419812776768710591670108148642844750810, 189187992620662930481983476012359292403998, 573409226327526684362114487387852400927492, 1737944022149435845036150193425076425233718, 5267528469100186080474912858807299167994915, 15965333646629475214731707491556164621480452, 48389274010272327826015760122646184198166796, 146662881657693877847234984806819199835036347, 444520015976525375964510783876742779154680757, 1347294164483672265081407271280381735326563565, 4083509179365312042620574956666103443769326617, 12376693715103556788071422317857288186871964030, 37512477770722827275164887057675807757709884757, 113696437909080164965352480316676027570831545567, 344602136713621660902503807674958465521545298382
Generating function in Maple syntax:
-(x^3-2*x^2+3*x-1)/(x^5-x^4-3*x^3+4*x^2-4*x+1)
Generating function in latex syntax:
-\frac{x^{3}-2 x^{2}+3 x -1}{x^{5}-x^{4}-3 x^{3}+4 x^{2}-4 x +1}
Generating function in sympy syntax:
(-x**3 + 2*x**2 - 3*x + 1)/(x**5 - x**4 - 3*x**3 + 4*x**2 - 4*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^5-x^4-3*x^3+4*x^2-4*x+1)*F(x)+x^3-2*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}-x^{4}-3 x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)+x^{3}-2 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
-1203/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+3)-1203/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+3)-1203/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+3)-1203/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+3)-1203/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+3)-211/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+2)-211/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+2)-211/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+2)-211/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+2)-211/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+2)+139/1801*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+1)+139/1801*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+1)+139/1801*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+1)+139/1801*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+1)+139/1801*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+1)+371/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n-1)+371/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n-1)+371/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n-1)+371/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n-1)+371/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n-1)+7863/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n)+7863/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n)+7863/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n)+7863/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n)+7863/46826*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n)
Explicit closed form in latex syntax:
-\frac{1203 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{46826}-\frac{1203 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{46826}-\frac{1203 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{46826}-\frac{1203 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{46826}-\frac{1203 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{46826}-\frac{211 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{46826}-\frac{211 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{46826}-\frac{211 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{46826}-\frac{211 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{46826}-\frac{211 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{46826}+\frac{139 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{1801}+\frac{139 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{1801}+\frac{139 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{1801}+\frac{139 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{1801}+\frac{139 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{1801}+\frac{371 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{23413}+\frac{371 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{23413}+\frac{371 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{23413}+\frac{371 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{23413}+\frac{371 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{23413}+\frac{7863 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{46826}+\frac{7863 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{46826}+\frac{7863 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{46826}+\frac{7863 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{46826}+\frac{7863 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{46826}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+5) = -a(n)+a(n+1)+3*a(n+2)-4*a(n+3)+4*a(n+4), 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 +5\right) = -a \! \left(n \right)+a \! \left(n +1\right)+3 a \! \left(n +2\right)-4 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15415/
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[20,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[11,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[16,x]
F[19,x] = F[13,x]*F[4,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[15,x]+F[22,x]+F[86,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[38,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[1,x]+F[4,x]
F[28,x] = F[15,x]+F[29,x]+F[36,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[17,x]+F[31,x]
F[31,x] = F[32,x]+F[34,x]
F[32,x] = F[15,x]+F[16,x]+F[33,x]
F[33,x] = F[11,x]*F[4,x]
F[34,x] = F[35,x]
F[35,x] = F[31,x]*F[4,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[11,x]
F[38,x] = F[39,x]+F[44,x]
F[39,x] = F[15,x]+F[22,x]+F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[2,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[2,x]*F[4,x]
F[44,x] = F[15,x]+F[45,x]+F[54,x]+F[84,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[4,x]
F[50,x] = F[51,x]
F[51,x] = 2*F[15,x]+F[45,x]+F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[42,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[77,x]
F[56,x] = F[57,x]+F[75,x]
F[57,x] = F[15,x]+F[43,x]+F[58,x]
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[15,x]+F[29,x]+F[33,x]
F[62,x] = F[57,x]+F[63,x]
F[63,x] = F[15,x]+F[54,x]+F[64,x]+F[71,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[68,x]
F[66,x] = F[67,x]
F[67,x] = x^2
F[68,x] = F[69,x]
F[69,x] = 2*F[15,x]+F[64,x]+F[70,x]
F[70,x] = F[4,x]*F[42,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[2,x]+F[72,x]
F[75,x] = F[76,x]
F[76,x] = F[4,x]*F[56,x]
F[77,x] = F[78,x]+F[82,x]
F[78,x] = 2*F[15,x]+F[71,x]+F[79,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[57,x]+F[81,x]
F[81,x] = F[79,x]
F[82,x] = F[83,x]
F[83,x] = F[4,x]*F[77,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[72,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[88,x]+F[91,x]
F[88,x] = F[2,x]+F[89,x]
F[89,x] = F[15,x]+F[58,x]+F[90,x]
F[90,x] = F[4,x]*F[88,x]
F[91,x] = F[72,x]+F[92,x]
F[92,x] = 2*F[15,x]+F[79,x]+F[93,x]
F[93,x] = F[4,x]*F[91,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_{20}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{86}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{38}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \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_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{36}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{17}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{11}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{39}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{2}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{54}\! \left(x \right)+F_{84}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \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_{4}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{45}\! \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_{42}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{77}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{75}\! \left(x \right)
F_{57}\! \left(x \right) = F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{58}\! \left(x \right)
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_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)
F_{62}\! \left(x \right) = F_{57}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{15}\! \left(x \right)+F_{54}\! \left(x \right)+F_{64}\! \left(x \right)+F_{71}\! \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_{68}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = x^{2}
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{64}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{2}\! \left(x \right)+F_{72}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{82}\! \left(x \right)
F_{78}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{71}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{57}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{79}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{72}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)+F_{91}\! \left(x \right)
F_{88}\! \left(x \right) = F_{2}\! \left(x \right)+F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{91}\! \left(x \right) = F_{72}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{79}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{4}\! \left(x \right) F_{91}\! \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_20(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_11(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_13(x)*F_4(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_15(x) + F_22(x) + F_86(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_38(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_1(x) + F_4(x))
Eq(F_28(x), F_15(x) + F_29(x) + F_36(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_17(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_34(x))
Eq(F_32(x), F_15(x) + F_16(x) + F_33(x))
Eq(F_33(x), F_11(x)*F_4(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_31(x)*F_4(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_11(x))
Eq(F_38(x), F_39(x) + F_44(x))
Eq(F_39(x), F_15(x) + F_22(x) + F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_2(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_2(x)*F_4(x))
Eq(F_44(x), F_15(x) + F_45(x) + F_54(x) + F_84(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_4(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), 2*F_15(x) + F_45(x) + F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_42(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_77(x))
Eq(F_56(x), F_57(x) + F_75(x))
Eq(F_57(x), F_15(x) + F_43(x) + F_58(x))
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_15(x) + F_29(x) + F_33(x))
Eq(F_62(x), F_57(x) + F_63(x))
Eq(F_63(x), F_15(x) + F_54(x) + F_64(x) + F_71(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_68(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), x**2)
Eq(F_68(x), F_69(x))
Eq(F_69(x), 2*F_15(x) + F_64(x) + F_70(x))
Eq(F_70(x), F_4(x)*F_42(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_2(x) + F_72(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_4(x)*F_56(x))
Eq(F_77(x), F_78(x) + F_82(x))
Eq(F_78(x), 2*F_15(x) + F_71(x) + F_79(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_57(x) + F_81(x))
Eq(F_81(x), F_79(x))
Eq(F_82(x), F_83(x))
Eq(F_83(x), F_4(x)*F_77(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_72(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_88(x) + F_91(x))
Eq(F_88(x), F_2(x) + F_89(x))
Eq(F_89(x), F_15(x) + F_58(x) + F_90(x))
Eq(F_90(x), F_4(x)*F_88(x))
Eq(F_91(x), F_72(x) + F_92(x))
Eq(F_92(x), 2*F_15(x) + F_79(x) + F_93(x))
Eq(F_93(x), F_4(x)*F_91(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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 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, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}], "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, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}], "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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 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