0123_0132_0321_1032_2130
Counting sequence:
1, 1, 2, 6, 19, 52, 128, 302, 703, 1613, 3651, 8183, 18207, 40270, 88633, 194303, 424572, 925237, 2011774, 4366071, 9460621, 20472647, 44253094, 95566020, 206213108, 444666644, 958302554, 2064229281, 4444593438, 9566470968, 20584476283, 44280809636, 95234736719, 204782722964, 440272198264, 946431967378, 2034259756251, 4371995337349, 9395406521163, 20189216773507, 43380644126587, 93207122693502, 200254445013729, 430227409293011, 924270952571944, 1985582638788909, 4265459981166694, 9162934192432531, 19683185225229001, 42281408031991895, 90823393528403822, 195092718543543320, 419063812652522096, 900151546867639900, 1933517017061960706, 4153150852415302477, 8920827392317764298, 19161549128432503640, 41158015667581948787, 88404996191996408632, 189888195447145285867, 407866538067602830284, 876066933389556753312, 1881723166568009896798, 4041787875815378151783, 8681419416573375756485, 18646935875001317129779, 40051962702825694943335, 86027989802725549338903, 184780204675793524056614, 396890636628671677907353, 852483614953324747669871, 1831053515293818727782436, 3932926873051438833810765, 8447546011392397542378382, 18144505679218746476137703, 38972620965506928371174693, 83709355625018415885986175, 179799429321814216639455782, 386191361304054719314212500, 829500678715643105565860124, 1781684859641768237987082988, 3826881310282948925204717386, 8219758572327111277756537105, 17655218677481801371089892214, 37921640353316819384528016432, 81451878990497386713133301563, 174950456930347550699605360212, 375775999751181567198115409559, 807129058354521056153169958508, 1733632003581635889625758228168, 3723667019074933142037391601826, 7998061738172350699568718681043, 17179030821310955316679830914653, 36898826910750390596299049391819, 79254960580551179833209030881587, 170231664147796347428323521874339, 365640447027108857419303184816470, 785358799924152450827239973507761, 1686871475878959422626349692157667, 3623229739335525569678655916980448
Generating function in Maple syntax:
-(2*x^12+5*x^11+5*x^10-x^9-8*x^8-9*x^7+6*x^5+6*x^4+x^3-2*x+1)/(x-1)/(x^2+1)/(x^3+2*x^2+x-1)/(x^3+x^2+x-1)
Generating function in latex syntax:
-\frac{2 x^{12}+5 x^{11}+5 x^{10}-x^{9}-8 x^{8}-9 x^{7}+6 x^{5}+6 x^{4}+x^{3}-2 x +1}{\left(x -1\right) \left(x^{2}+1\right) \left(x^{3}+2 x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(-2*x**12 - 5*x**11 - 5*x**10 + x**9 + 8*x**8 + 9*x**7 - 6*x**5 - 6*x**4 - x**3 + 2*x - 1)/((x - 1)*(x**2 + 1)*(x**3 + x**2 + x - 1)*(x**3 + 2*x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2+1)*(x^3+2*x^2+x-1)*(x^3+x^2+x-1)*F(x)+2*x^12+5*x^11+5*x^10-x^9-8*x^8-9*x^7+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(x^{2}+1\right) \left(x^{3}+2 x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+2 x^{12}+5 x^{11}+5 x^{10}-x^{9}-8 x^{8}-9 x^{7}+6 x^{5}+6 x^{4}+x^{3}-2 x +1 = 0
Explicit closed form in Maple syntax:
20203/4092*Sum(_alpha^(-n+7),_alpha = RootOf(_Z^9+2*_Z^8+2*_Z^7-_Z^6-2*_Z^5-3*_Z^4-_Z^2+3*_Z-1))+301633/24552*Sum(_alpha^(-n+6),_alpha = RootOf(_Z^9+2*_Z^8+2*_Z^7-_Z^6-2*_Z^5-3*_Z^4-_Z^2+3*_Z-1))+194827/12276*Sum(_alpha^(-n+5),_alpha = RootOf(_Z^9+2*_Z^8+2*_Z^7-_Z^6-2*_Z^5-3*_Z^4-_Z^2+3*_Z-1))+74821/24552*Sum(_alpha^(-n+4),_alpha = RootOf(_Z^9+2*_Z^8+2*_Z^7-_Z^6-2*_Z^5-3*_Z^4-_Z^2+3*_Z-1))-64663/8184*Sum(_alpha^(3-n),_alpha = RootOf(_Z^9+2*_Z^8+2*_Z^7-_Z^6-2*_Z^5-3*_Z^4-_Z^2+3*_Z-1))-50037/2728*Sum(_alpha^(2-n),_alpha = RootOf(_Z^9+2*_Z^8+2*_Z^7-_Z^6-2*_Z^5-3*_Z^4-_Z^2+3*_Z-1))-38113/4092*Sum(_alpha^(1-n),_alpha = RootOf(_Z^9+2*_Z^8+2*_Z^7-_Z^6-2*_Z^5-3*_Z^4-_Z^2+3*_Z-1))-77917/8184*Sum(_alpha^(-n),_alpha = RootOf(_Z^9+2*_Z^8+2*_Z^7-_Z^6-2*_Z^5-3*_Z^4-_Z^2+3*_Z-1))+231701/24552*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^9+2*_Z^8+2*_Z^7-_Z^6-2*_Z^5-3*_Z^4-_Z^2+3*_Z-1))-piecewise(n = 0,1,n = 1,-1,n = 2,1,n = 3,2,0)
Explicit closed form in latex syntax:
\frac{20203 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \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 +7}\right)}{4092}+\frac{301633 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \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)}{24552}+\frac{194827 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \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)}{12276}+\frac{74821 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \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)}{24552}-\frac{64663 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \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)}{8184}-\frac{50037 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \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)}{2728}-\frac{38113 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{4092}-\frac{77917 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \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)}{8184}+\frac{231701 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \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)}{24552}-\left(\left\{\begin{array}{cc}1 & n =0 \\ -1 & n =1 \\ 1 & 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) = 52
a(6) = 128
a(7) = 302
a(8) = 703
a(9) = 1613
a(10) = 3651
a(11) = 8183
a(12) = 18207
a(n+8) = -a(n)-3*a(n+1)-5*a(n+2)-4*a(n+3)-2*a(n+4)+a(n+5)+a(n+6)+2*a(n+7)+6, n >= 13
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) = 52
a \! \left(6\right) = 128
a \! \left(7\right) = 302
a \! \left(8\right) = 703
a \! \left(9\right) = 1613
a \! \left(10\right) = 3651
a \! \left(11\right) = 8183
a \! \left(12\right) = 18207
a \! \left(n +8\right) = -a \! \left(n \right)-3 a \! \left(n +1\right)-5 a \! \left(n +2\right)-4 a \! \left(n +3\right)-2 a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right)+2 a \! \left(n +7\right)+6, \quad n \geq 13
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16226/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[4,x]
F[18,x] = F[12,x]*F[4,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,x]+F[21,x]+F[72,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[30,x]
F[23,x] = F[12,x]+F[24,x]
F[24,x] = F[15,x]+F[25,x]+F[29,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[16,x]
F[29,x] = F[11,x]*F[4,x]
F[30,x] = F[31,x]+F[71,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[36,x]
F[34,x] = F[12,x]+F[35,x]
F[35,x] = F[25,x]
F[36,x] = F[31,x]+F[37,x]
F[37,x] = 2*F[15,x]+F[38,x]+F[49,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[44,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[16,x]
F[42,x] = F[43,x]
F[43,x] = F[27,x]*F[4,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[41,x]
F[48,x] = F[45,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[68,x]
F[51,x] = F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[57,x]
F[55,x] = F[4,x]+F[56,x]
F[56,x] = F[25,x]
F[57,x] = F[52,x]+F[58,x]
F[58,x] = 2*F[15,x]+F[49,x]+F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[63,x]
F[61,x] = F[28,x]+F[62,x]
F[62,x] = F[43,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[67,x]
F[66,x] = F[28,x]
F[67,x] = F[64,x]
F[68,x] = F[69,x]
F[69,x] = 2*F[15,x]+F[59,x]+F[70,x]
F[70,x] = F[4,x]*F[51,x]
F[71,x] = 2*F[15,x]+F[38,x]+F[49,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[74,x]+F[83,x]
F[74,x] = F[2,x]+F[75,x]
F[75,x] = F[15,x]+F[76,x]+F[82,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[78,x]+F[80,x]
F[78,x] = F[4,x]+F[79,x]
F[79,x] = F[15,x]+F[18,x]+F[25,x]
F[80,x] = F[52,x]+F[81,x]
F[81,x] = 2*F[15,x]+F[49,x]+F[59,x]
F[82,x] = F[2,x]*F[4,x]
F[83,x] = F[84,x]+F[86,x]
F[84,x] = F[15,x]+F[21,x]+F[85,x]
F[85,x] = F[4,x]*F[74,x]
F[86,x] = F[15,x]+F[70,x]+F[87,x]+F[92,x]
F[87,x] = F[4,x]*F[88,x]
F[88,x] = F[89,x]+F[91,x]
F[89,x] = F[28,x]+F[90,x]
F[90,x] = F[43,x]
F[91,x] = F[64,x]
F[92,x] = F[4,x]*F[84,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \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_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{72}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{71}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{49}\! \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_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{16}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \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_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{41}\! \left(x \right)
F_{48}\! \left(x \right) = F_{45}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{68}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \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_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{25}\! \left(x \right)
F_{57}\! \left(x \right) = F_{52}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{28}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{43}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{28}\! \left(x \right)
F_{67}\! \left(x \right) = F_{64}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{59}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{71}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{49}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{83}\! \left(x \right)
F_{74}\! \left(x \right) = F_{2}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{15}\! \left(x \right)+F_{76}\! \left(x \right)+F_{82}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{80}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{80}\! \left(x \right) = F_{52}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{59}\! \left(x \right)
F_{82}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{86}\! \left(x \right)
F_{84}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{86}\! \left(x \right) = F_{15}\! \left(x \right)+F_{70}\! \left(x \right)+F_{87}\! \left(x \right)+F_{92}\! \left(x \right)
F_{87}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{91}\! \left(x \right)
F_{89}\! \left(x \right) = F_{28}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{43}\! \left(x \right)
F_{91}\! \left(x \right) = F_{64}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{84}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_12(x)*F_4(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_21(x) + F_72(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_30(x))
Eq(F_23(x), F_12(x) + F_24(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_29(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_16(x))
Eq(F_29(x), F_11(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_71(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_36(x))
Eq(F_34(x), F_12(x) + F_35(x))
Eq(F_35(x), F_25(x))
Eq(F_36(x), F_31(x) + F_37(x))
Eq(F_37(x), 2*F_15(x) + F_38(x) + F_49(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_44(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_16(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_27(x)*F_4(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_41(x))
Eq(F_48(x), F_45(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_68(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_57(x))
Eq(F_55(x), F_4(x) + F_56(x))
Eq(F_56(x), F_25(x))
Eq(F_57(x), F_52(x) + F_58(x))
Eq(F_58(x), 2*F_15(x) + F_49(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_63(x))
Eq(F_61(x), F_28(x) + F_62(x))
Eq(F_62(x), F_43(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_67(x))
Eq(F_66(x), F_28(x))
Eq(F_67(x), F_64(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), 2*F_15(x) + F_59(x) + F_70(x))
Eq(F_70(x), F_4(x)*F_51(x))
Eq(F_71(x), 2*F_15(x) + F_38(x) + F_49(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_83(x))
Eq(F_74(x), F_2(x) + F_75(x))
Eq(F_75(x), F_15(x) + F_76(x) + F_82(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_80(x))
Eq(F_78(x), F_4(x) + F_79(x))
Eq(F_79(x), F_15(x) + F_18(x) + F_25(x))
Eq(F_80(x), F_52(x) + F_81(x))
Eq(F_81(x), 2*F_15(x) + F_49(x) + F_59(x))
Eq(F_82(x), F_2(x)*F_4(x))
Eq(F_83(x), F_84(x) + F_86(x))
Eq(F_84(x), F_15(x) + F_21(x) + F_85(x))
Eq(F_85(x), F_4(x)*F_74(x))
Eq(F_86(x), F_15(x) + F_70(x) + F_87(x) + F_92(x))
Eq(F_87(x), F_4(x)*F_88(x))
Eq(F_88(x), F_89(x) + F_91(x))
Eq(F_89(x), F_28(x) + F_90(x))
Eq(F_90(x), F_43(x))
Eq(F_91(x), F_64(x))
Eq(F_92(x), F_4(x)*F_84(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, 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, 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, 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, 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, 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, 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": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"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, 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, 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": [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": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 2], [1, 0]]}, {"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, 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, 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": [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": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[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, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"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, 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, 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": [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": [[{"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], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"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, 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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 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