0123_0231_0321_1032_2130
Counting sequence:
1, 1, 2, 6, 19, 51, 124, 290, 671, 1528, 3428, 7619, 16810, 36854, 80383, 174604, 377968, 815839, 1756742, 3775138, 8098667, 17348520, 37117092, 79328099, 169390146, 361421078, 770638679, 1642255516, 3497984600, 7447523935, 15850688462, 33724716434, 71735001619, 152550137848, 324344547724, 689485294803, 1465473888058, 3114402092198, 6617934143231, 14061380383868, 29874244286448, 63464999200655, 134816878182342, 286372012222994, 608269970579435, 1291946221908728, 2743955936338404, 5827691164431763, 12376686973706146, 26284656946211174, 55820225141576087, 118542307363720172, 251737950856485432, 534587008926910191, 1135228410836129838, 2410704039288077538, 5119185845348896691, 10870630688947720360, 23083722978072222252, 49017888420263645667, 104088150959210713146, 221027449733292800214, 469342187985205171103, 996624690485536778156, 2116277121379647543280, 4493786467330198842495, 9542264741211337519238, 20262346649383829385282, 43025643160714809870123, 91361756032553593788552, 193999682189642493852132, 411943046455648954539651, 874727879665639482291394, 1857412844027931183467254, 3944061203693470948867479, 8374880316769358190259196, 17783341448826917612586456, 37761387781530723780045247, 80183013089557269709620046, 170261581622023105988564210, 361535409130803833127920403, 767688286491527631672230552, 1630117472220267657310615180, 3461408177347257488550798771, 7349988559113296543487210746, 15607036796489836840478275398, 33140129152774792695523367167, 70370055981129147318412784028, 149424413444695917750511873456, 317289127980920053847742868143, 673734512508518702019921841926, 1430613684412335177074768498482, 3037777312314992693461064020523, 6450442016135453394904075407448, 13696922598335555021426552247460, 29084159254941220216077102496819, 61757543771665872660413689011106, 131136475040096272188878644171206, 278456261211366619108880805239127, 591276287043763368561228547758796, 1255520852095338655143489294616888
Generating function in Maple syntax:
-(4*x^11+5*x^10+3*x^9-6*x^8-10*x^7-2*x^6+6*x^5+6*x^4+x^3-2*x+1)/(x-1)/(2*x^4+2*x^3+x^2+x-1)/(x^3+x^2+x-1)
Generating function in latex syntax:
-\frac{4 x^{11}+5 x^{10}+3 x^{9}-6 x^{8}-10 x^{7}-2 x^{6}+6 x^{5}+6 x^{4}+x^{3}-2 x +1}{\left(x -1\right) \left(2 x^{4}+2 x^{3}+x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(-4*x**11 - 5*x**10 - 3*x**9 + 6*x**8 + 10*x**7 + 2*x**6 - 6*x**5 - 6*x**4 - x**3 + 2*x - 1)/((x - 1)*(x**3 + x**2 + x - 1)*(2*x**4 + 2*x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x^4+2*x^3+x^2+x-1)*(x^3+x^2+x-1)*F(x)+4*x^11+5*x^10+3*x^9-6*x^8-10*x^7-2*x^6+6*x^5+6*x^4+x^3-2*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x^{4}+2 x^{3}+x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+4 x^{11}+5 x^{10}+3 x^{9}-6 x^{8}-10 x^{7}-2 x^{6}+6 x^{5}+6 x^{4}+x^{3}-2 x +1 = 0
Explicit closed form in Maple syntax:
5781104/480975*Sum(_alpha^(-n+6),_alpha = RootOf(2*_Z^8+2*_Z^7+_Z^6-3*_Z^5-3*_Z^4-_Z^2+3*_Z-1))+8669303/480975*Sum(_alpha^(-n+5),_alpha = RootOf(2*_Z^8+2*_Z^7+_Z^6-3*_Z^5-3*_Z^4-_Z^2+3*_Z-1))+974791/64130*Sum(_alpha^(-n+4),_alpha = RootOf(2*_Z^8+2*_Z^7+_Z^6-3*_Z^5-3*_Z^4-_Z^2+3*_Z-1))-4735211/480975*Sum(_alpha^(3-n),_alpha = RootOf(2*_Z^8+2*_Z^7+_Z^6-3*_Z^5-3*_Z^4-_Z^2+3*_Z-1))-1995629/87450*Sum(_alpha^(2-n),_alpha = RootOf(2*_Z^8+2*_Z^7+_Z^6-3*_Z^5-3*_Z^4-_Z^2+3*_Z-1))-1008349/87450*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^8+2*_Z^7+_Z^6-3*_Z^5-3*_Z^4-_Z^2+3*_Z-1))-5848969/480975*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^8+2*_Z^7+_Z^6-3*_Z^5-3*_Z^4-_Z^2+3*_Z-1))+11266609/961950*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^8+2*_Z^7+_Z^6-3*_Z^5-3*_Z^4-_Z^2+3*_Z-1))-piecewise(n = 0,-1/4,n = 2,1/2,n = 3,2,0)
Explicit closed form in latex syntax:
\frac{5781104 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +6}\right)}{480975}+\frac{8669303 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +5}\right)}{480975}+\frac{974791 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{64130}-\frac{4735211 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{3-n}\right)}{480975}-\frac{1995629 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{2-n}\right)}{87450}-\frac{1008349 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{87450}-\frac{5848969 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{480975}+\frac{11266609 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{961950}-\left(\left\{\begin{array}{cc}-\frac{1}{4} & n =0 \\ \frac{1}{2} & n =2 \\ 2 & n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 124
a(7) = 290
a(8) = 671
a(9) = 1528
a(10) = 3428
a(11) = 7619
a(n+7) = -2*a(n)-4*a(n+1)-5*a(n+2)-2*a(n+3)+a(n+4)+a(n+5)+2*a(n+6)+6, n >= 12
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) = 51
a \! \left(6\right) = 124
a \! \left(7\right) = 290
a \! \left(8\right) = 671
a \! \left(9\right) = 1528
a \! \left(10\right) = 3428
a \! \left(11\right) = 7619
a \! \left(n +7\right) = -2 a \! \left(n \right)-4 a \! \left(n +1\right)-5 a \! \left(n +2\right)-2 a \! \left(n +3\right)+a \! \left(n +4\right)+a \! \left(n +5\right)+2 a \! \left(n +6\right)+6, \quad n \geq 12
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17078/
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[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[4,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[88,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[36,x]
F[25,x] = F[15,x]+F[26,x]
F[26,x] = F[22,x]+F[27,x]+F[34,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[30,x]+F[4,x]
F[30,x] = F[31,x]
F[31,x] = x^2
F[32,x] = F[33,x]
F[33,x] = F[18,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[15,x]+F[33,x]
F[36,x] = F[37,x]+F[83,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[46,x]
F[40,x] = F[15,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[4,x]
F[45,x] = F[33,x]
F[46,x] = F[37,x]+F[47,x]
F[47,x] = 2*F[22,x]+F[48,x]+F[60,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[55,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[18,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[33,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[51,x]
F[59,x] = F[56,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[79,x]
F[62,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[68,x]
F[66,x] = F[4,x]+F[67,x]
F[67,x] = F[42,x]
F[68,x] = F[63,x]+F[69,x]
F[69,x] = 2*F[22,x]+F[60,x]+F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]+F[74,x]
F[72,x] = F[33,x]+F[73,x]
F[73,x] = F[53,x]
F[74,x] = F[75,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[77,x]+F[78,x]
F[77,x] = F[33,x]
F[78,x] = F[75,x]
F[79,x] = F[80,x]
F[80,x] = 2*F[22,x]+F[70,x]+F[81,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[63,x]
F[83,x] = 2*F[22,x]+F[48,x]+F[84,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[86,x]+F[87,x]
F[86,x] = F[63,x]
F[87,x] = F[80,x]
F[88,x] = F[4,x]*F[89,x]
F[89,x] = F[103,x]+F[90,x]
F[90,x] = F[2,x]+F[91,x]
F[91,x] = F[22,x]+F[92,x]+F[99,x]
F[92,x] = F[4,x]*F[93,x]
F[93,x] = F[94,x]+F[97,x]
F[94,x] = F[4,x]+F[95,x]
F[95,x] = F[22,x]+F[27,x]+F[96,x]
F[96,x] = F[15,x]*F[4,x]
F[97,x] = F[63,x]+F[98,x]
F[98,x] = 2*F[22,x]+F[70,x]+F[84,x]
F[99,x] = F[100,x]*F[4,x]
F[100,x] = F[101,x]+F[2,x]
F[101,x] = F[102,x]+F[22,x]+F[92,x]
F[102,x] = F[2,x]*F[4,x]
F[103,x] = F[104,x]+F[106,x]
F[104,x] = F[105,x]+F[22,x]+F[23,x]
F[105,x] = F[100,x]*F[4,x]
F[106,x] = 2*F[22,x]+F[107,x]+F[81,x]
F[107,x] = F[108,x]*F[4,x]
F[108,x] = F[109,x]+F[111,x]
F[109,x] = F[110,x]+F[33,x]
F[110,x] = F[53,x]
F[111,x] = F[75,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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \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_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{88}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{36}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = x^{2}
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{83}\! \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_{40}\! \left(x \right)+F_{46}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{33}\! \left(x \right)
F_{46}\! \left(x \right) = F_{37}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{60}\! \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_{55}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{18}\! \left(x \right)
F_{52}\! \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_{33}\! \left(x \right)
F_{55}\! \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_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{51}\! \left(x \right)
F_{59}\! \left(x \right) = F_{56}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{79}\! \left(x \right)
F_{62}\! \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_{68}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{42}\! \left(x \right)
F_{68}\! \left(x \right) = F_{63}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{60}\! \left(x \right)+F_{70}\! \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_{33}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{53}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{78}\! \left(x \right)
F_{77}\! \left(x \right) = F_{33}\! \left(x \right)
F_{78}\! \left(x \right) = F_{75}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{70}\! \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_{63}\! \left(x \right)
F_{83}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)+F_{87}\! \left(x \right)
F_{86}\! \left(x \right) = F_{63}\! \left(x \right)
F_{87}\! \left(x \right) = F_{80}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{103}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{2}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{22}\! \left(x \right)+F_{92}\! \left(x \right)+F_{99}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)+F_{97}\! \left(x \right)
F_{94}\! \left(x \right) = F_{4}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{97}\! \left(x \right) = F_{63}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{70}\! \left(x \right)+F_{84}\! \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_{2}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{22}\! \left(x \right)+F_{92}\! \left(x \right)
F_{102}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{106}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{105}\! \left(x \right) = F_{100}\! \left(x \right) F_{4}\! \left(x \right)
F_{106}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{107}\! \left(x \right)+F_{81}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{4}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{111}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)+F_{33}\! \left(x \right)
F_{110}\! \left(x \right) = F_{53}\! \left(x \right)
F_{111}\! \left(x \right) = F_{75}\! \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_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_4(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_88(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_36(x))
Eq(F_25(x), F_15(x) + F_26(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_34(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_30(x) + F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), x**2)
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_18(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_15(x) + F_33(x))
Eq(F_36(x), F_37(x) + F_83(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_46(x))
Eq(F_40(x), F_15(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_4(x))
Eq(F_45(x), F_33(x))
Eq(F_46(x), F_37(x) + F_47(x))
Eq(F_47(x), 2*F_22(x) + F_48(x) + F_60(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_55(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_18(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_33(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_51(x))
Eq(F_59(x), F_56(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_79(x))
Eq(F_62(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_68(x))
Eq(F_66(x), F_4(x) + F_67(x))
Eq(F_67(x), F_42(x))
Eq(F_68(x), F_63(x) + F_69(x))
Eq(F_69(x), 2*F_22(x) + F_60(x) + F_70(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_33(x) + F_73(x))
Eq(F_73(x), F_53(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_77(x) + F_78(x))
Eq(F_77(x), F_33(x))
Eq(F_78(x), F_75(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), 2*F_22(x) + F_70(x) + F_81(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_63(x))
Eq(F_83(x), 2*F_22(x) + F_48(x) + F_84(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_86(x) + F_87(x))
Eq(F_86(x), F_63(x))
Eq(F_87(x), F_80(x))
Eq(F_88(x), F_4(x)*F_89(x))
Eq(F_89(x), F_103(x) + F_90(x))
Eq(F_90(x), F_2(x) + F_91(x))
Eq(F_91(x), F_22(x) + F_92(x) + F_99(x))
Eq(F_92(x), F_4(x)*F_93(x))
Eq(F_93(x), F_94(x) + F_97(x))
Eq(F_94(x), F_4(x) + F_95(x))
Eq(F_95(x), F_22(x) + F_27(x) + F_96(x))
Eq(F_96(x), F_15(x)*F_4(x))
Eq(F_97(x), F_63(x) + F_98(x))
Eq(F_98(x), 2*F_22(x) + F_70(x) + F_84(x))
Eq(F_99(x), F_100(x)*F_4(x))
Eq(F_100(x), F_101(x) + F_2(x))
Eq(F_101(x), F_102(x) + F_22(x) + F_92(x))
Eq(F_102(x), F_2(x)*F_4(x))
Eq(F_103(x), F_104(x) + F_106(x))
Eq(F_104(x), F_105(x) + F_22(x) + F_23(x))
Eq(F_105(x), F_100(x)*F_4(x))
Eq(F_106(x), F_107(x) + 2*F_22(x) + F_81(x))
Eq(F_107(x), F_108(x)*F_4(x))
Eq(F_108(x), F_109(x) + F_111(x))
Eq(F_109(x), F_110(x) + F_33(x))
Eq(F_110(x), F_53(x))
Eq(F_111(x), F_75(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": [0, 3, 2, 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, 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": [0, 3, 2, 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": "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": [0, 3, 2, 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": []}, "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, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 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, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [0, 1], [0, 1], [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, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 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, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"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": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 2], [1, 2], [1, 2], [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": 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