0123_0321_1032_1302_3102
Counting sequence:
1, 1, 2, 6, 19, 50, 123, 299, 714, 1668, 3844, 8782, 19918, 44896, 100713, 225083, 501505, 1114592, 2472107, 5473841, 12103689, 26732971, 58988338, 130060313, 286576534, 631104245, 1389202547, 3056797455, 6724058010, 14787080914, 32511604735, 71468546293, 157081524162, 345207129233, 758556579197, 1666699726850, 3661796896435, 8044590571270, 17672210605592, 38820288637723, 85272833405355, 187304959193467, 411411740102871, 903638500398679, 1984746000684667, 4359217950830674, 9574293501085882, 21028112375354134, 46183834589150321, 101432344178489632, 222771810320438886, 489262289319612798, 1074536957861710476, 2359931432031932401, 5182938829494896694, 11382867192158735835, 24999212168158784142, 54903520815252794076, 120579482286399396504, 264817137503619102821, 581591834316979112669, 1277291816864966075698, 2805185838580784455124, 6160739938651587376423, 13530189821766284317432, 29714930157332513830358, 65259746113974534037163, 143323007699531024562396, 314764928022083325646569, 691284255788410646320455, 1518192765874013699795522, 3334241897119405869181566, 7322632462246554092158726, 16081898730777283681387862, 35318916431059861806552066, 77567070074135758813420150, 170352054815845271567012159, 374125528392875208951223111, 821650816462954206339913021, 1804501399774424501025737198, 3963028013246439492051475057, 8703562455358324840894199409, 19114676339687508897307276885, 41979459206195818496046118489, 92194863128153837387702169504, 202477422087709240827140861313, 444678853398249715273500024542, 976599162922819014264663743115, 2144797106299659262522563676141, 4710381468140625555411239393231, 10344891549733539660992797106754, 22719344799748229936277533452680, 49895992024588381662775955251629, 109581065585801988867289059870593, 240660810962616440548115077318746, 528536800590242113988231295966207, 1160767089428309726597578802717143, 2549264748235169946733287514054182, 5598669024701008599785066087692763, 12295739325249258866381076095914168, 27003776210911681002321147852380634
Generating function in Maple syntax:
(x^9+x^8+x^7-x^6-5*x^5-5*x^4-x^3+2*x-1)/(x-1)/(x^3+x^2+x-1)/(x^5+3*x^4+2*x^3+x^2+x-1)
Generating function in latex syntax:
\frac{x^{9}+x^{8}+x^{7}-x^{6}-5 x^{5}-5 x^{4}-x^{3}+2 x -1}{\left(x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(x**9 + x**8 + x**7 - x**6 - 5*x**5 - 5*x**4 - x**3 + 2*x - 1)/((x - 1)*(x**3 + x**2 + x - 1)*(x**5 + 3*x**4 + 2*x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^3+x^2+x-1)*(x^5+3*x^4+2*x^3+x^2+x-1)*F(x)-x^9-x^8-x^7+x^6+5*x^5+5*x^4+x^3-2*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)-x^{9}-x^{8}-x^{7}+x^{6}+5 x^{5}+5 x^{4}+x^{3}-2 x +1 = 0
Explicit closed form in Maple syntax:
4754285889/1433463493*Sum(_alpha^(-n+7),_alpha = RootOf(_Z^9+3*_Z^8+2*_Z^7-_Z^6-4*_Z^5-2*_Z^4-_Z^2+3*_Z-1))+3029249873/260629726*Sum(_alpha^(-n+6),_alpha = RootOf(_Z^9+3*_Z^8+2*_Z^7-_Z^6-4*_Z^5-2*_Z^4-_Z^2+3*_Z-1))+36048502043/2866926986*Sum(_alpha^(-n+5),_alpha = RootOf(_Z^9+3*_Z^8+2*_Z^7-_Z^6-4*_Z^5-2*_Z^4-_Z^2+3*_Z-1))+8926154883/2866926986*Sum(_alpha^(-n+4),_alpha = RootOf(_Z^9+3*_Z^8+2*_Z^7-_Z^6-4*_Z^5-2*_Z^4-_Z^2+3*_Z-1))-16625828510/1433463493*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^9+3*_Z^8+2*_Z^7-_Z^6-4*_Z^5-2*_Z^4-_Z^2+3*_Z-1))-35308684165/2866926986*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^9+3*_Z^8+2*_Z^7-_Z^6-4*_Z^5-2*_Z^4-_Z^2+3*_Z-1))-8991656306/1433463493*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^9+3*_Z^8+2*_Z^7-_Z^6-4*_Z^5-2*_Z^4-_Z^2+3*_Z-1))-18112488371/2866926986*Sum(_alpha^(-n),_alpha = RootOf(_Z^9+3*_Z^8+2*_Z^7-_Z^6-4*_Z^5-2*_Z^4-_Z^2+3*_Z-1))+18489408853/2866926986*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^9+3*_Z^8+2*_Z^7-_Z^6-4*_Z^5-2*_Z^4-_Z^2+3*_Z-1))+piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
\frac{4754285889 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+3 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-4 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +7}\right)}{1433463493}+\frac{3029249873 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+3 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-4 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +6}\right)}{260629726}+\frac{36048502043 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+3 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-4 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +5}\right)}{2866926986}+\frac{8926154883 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+3 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-4 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{2866926986}-\frac{16625828510 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+3 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-4 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{1433463493}-\frac{35308684165 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+3 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-4 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{2866926986}-\frac{8991656306 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+3 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-4 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{1433463493}-\frac{18112488371 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+3 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-4 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{2866926986}+\frac{18489408853 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{9}+3 \textit{\_Z}^{8}+2 \textit{\_Z}^{7}-\textit{\_Z}^{6}-4 \textit{\_Z}^{5}-2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{2866926986}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 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) = 50
a(6) = 123
a(7) = 299
a(8) = 714
a(9) = 1668
a(n+8) = -a(n)-4*a(n+1)-6*a(n+2)-5*a(n+3)-a(n+4)+a(n+5)+a(n+6)+2*a(n+7)+8, n >= 10
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) = 50
a \! \left(6\right) = 123
a \! \left(7\right) = 299
a \! \left(8\right) = 714
a \! \left(9\right) = 1668
a \! \left(n +8\right) = -a \! \left(n \right)-4 a \! \left(n +1\right)-6 a \! \left(n +2\right)-5 a \! \left(n +3\right)-a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right)+2 a \! \left(n +7\right)+8, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17725/
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[25,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[19,x]+F[23,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[4,x]
F[21,x] = F[22,x]
F[22,x] = x^2
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[15,x]
F[25,x] = F[26,x]+F[51,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[35,x]
F[29,x] = F[1,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[10,x]+F[33,x]
F[33,x] = F[15,x]+F[34,x]
F[34,x] = F[23,x]
F[35,x] = F[26,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[45,x]
F[39,x] = F[26,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[26,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[26,x]*F[4,x]
F[45,x] = F[46,x]+F[48,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[42,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[46,x]
F[51,x] = F[18,x]+F[52,x]+F[90,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[65,x]
F[54,x] = F[15,x]+F[55,x]
F[55,x] = F[18,x]+F[56,x]+F[63,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[60,x]
F[58,x] = F[4,x]+F[59,x]
F[59,x] = F[22,x]
F[60,x] = F[61,x]
F[61,x] = F[18,x]+F[19,x]+F[62,x]
F[62,x] = F[15,x]*F[4,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[15,x]+F[61,x]
F[65,x] = F[66,x]+F[87,x]
F[66,x] = F[18,x]+F[52,x]+F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[26,x]+F[69,x]
F[69,x] = F[18,x]+F[44,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[4,x]+F[73,x]
F[73,x] = F[18,x]+F[56,x]+F[62,x]
F[74,x] = F[69,x]+F[75,x]
F[75,x] = 2*F[18,x]+F[76,x]+F[86,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[78,x]+F[81,x]
F[78,x] = F[69,x]+F[79,x]
F[79,x] = F[80,x]
F[80,x] = F[4,x]*F[43,x]
F[81,x] = F[82,x]
F[82,x] = 2*F[18,x]+F[83,x]+F[86,x]
F[83,x] = F[4,x]*F[84,x]
F[84,x] = F[69,x]+F[85,x]
F[85,x] = F[80,x]
F[86,x] = F[4,x]*F[46,x]
F[87,x] = 2*F[18,x]+F[76,x]+F[88,x]
F[88,x] = F[4,x]*F[89,x]
F[89,x] = F[46,x]+F[82,x]
F[90,x] = F[4,x]*F[91,x]
F[91,x] = F[92,x]+F[94,x]
F[92,x] = F[26,x]+F[93,x]
F[93,x] = F[18,x]+F[41,x]+F[70,x]
F[94,x] = F[46,x]+F[95,x]
F[95,x] = 2*F[18,x]+F[49,x]+F[83,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_{25}\! \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_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \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_{22}\! \left(x \right) = x^{2}
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{51}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \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_{35}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{10}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{23}\! \left(x \right)
F_{35}\! \left(x \right) = F_{26}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{45}\! \left(x \right)
F_{39}\! \left(x \right) = F_{26}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{26}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{46}\! \left(x \right)
F_{51}\! \left(x \right) = F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{90}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{65}\! \left(x \right)
F_{54}\! \left(x \right) = F_{15}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{63}\! \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_{60}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{22}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{15}\! \left(x \right)+F_{61}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{87}\! \left(x \right)
F_{66}\! \left(x \right) = F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{26}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{18}\! \left(x \right)+F_{44}\! \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_{4}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{62}\! \left(x \right)
F_{74}\! \left(x \right) = F_{69}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{76}\! \left(x \right)+F_{86}\! \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_{81}\! \left(x \right)
F_{78}\! \left(x \right) = F_{69}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{83}\! \left(x \right)+F_{86}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{69}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{80}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{87}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{76}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{46}\! \left(x \right)+F_{82}\! \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_{26}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{70}\! \left(x \right)
F_{94}\! \left(x \right) = F_{46}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{83}\! \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_25(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) + F_19(x) + F_23(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), x**2)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_15(x))
Eq(F_25(x), F_26(x) + F_51(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_35(x))
Eq(F_29(x), F_1(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_10(x) + F_33(x))
Eq(F_33(x), F_15(x) + F_34(x))
Eq(F_34(x), F_23(x))
Eq(F_35(x), F_26(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_45(x))
Eq(F_39(x), F_26(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_26(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_26(x)*F_4(x))
Eq(F_45(x), F_46(x) + F_48(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_42(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_46(x))
Eq(F_51(x), F_18(x) + F_52(x) + F_90(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_65(x))
Eq(F_54(x), F_15(x) + F_55(x))
Eq(F_55(x), F_18(x) + F_56(x) + F_63(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_60(x))
Eq(F_58(x), F_4(x) + F_59(x))
Eq(F_59(x), F_22(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_18(x) + F_19(x) + F_62(x))
Eq(F_62(x), F_15(x)*F_4(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_15(x) + F_61(x))
Eq(F_65(x), F_66(x) + F_87(x))
Eq(F_66(x), F_18(x) + F_52(x) + F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_26(x) + F_69(x))
Eq(F_69(x), F_18(x) + F_44(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_4(x) + F_73(x))
Eq(F_73(x), F_18(x) + F_56(x) + F_62(x))
Eq(F_74(x), F_69(x) + F_75(x))
Eq(F_75(x), 2*F_18(x) + F_76(x) + F_86(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_81(x))
Eq(F_78(x), F_69(x) + F_79(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_4(x)*F_43(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), 2*F_18(x) + F_83(x) + F_86(x))
Eq(F_83(x), F_4(x)*F_84(x))
Eq(F_84(x), F_69(x) + F_85(x))
Eq(F_85(x), F_80(x))
Eq(F_86(x), F_4(x)*F_46(x))
Eq(F_87(x), 2*F_18(x) + F_76(x) + F_88(x))
Eq(F_88(x), F_4(x)*F_89(x))
Eq(F_89(x), F_46(x) + F_82(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_26(x) + F_93(x))
Eq(F_93(x), F_18(x) + F_41(x) + F_70(x))
Eq(F_94(x), F_46(x) + F_95(x))
Eq(F_95(x), 2*F_18(x) + F_49(x) + F_83(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, 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": [1, 3, 0, 2], "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, 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": [1, 3, 0, 2], "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, 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": [1, 3, 0, 2], "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, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"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, 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, 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, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 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, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"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, 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, 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, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [1, 3, 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": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 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, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"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, 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, 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, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"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, 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, 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, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": 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