0123_0321_1032_2130
Counting sequence:
1, 1, 2, 6, 20, 58, 145, 347, 818, 1911, 4400, 10027, 22698, 51110, 114557, 255818, 569616, 1265342, 2805323, 6209530, 13726547, 30310287, 66869101, 147412565, 324767407, 715129652, 1574016023, 3463193603, 7617520356, 16751021578, 36827969037, 80953946193, 177924012485, 391000961488, 859164892164, 1887721562436, 4147325814192, 9111130484195, 20014950816703, 43966156143484, 96575542750081, 212130443000442, 465937975629719, 1023397219058596, 2247774980628670, 4936909128582905, 10843067434330030, 23814680895517992, 52303852109497114, 114873402857245109, 252291548906624654, 554094382799752691, 1216922644675347609, 2672641293413330915, 5869716147435527257, 12891173599149879983, 28311763323496281608, 62178555981505641633, 136556921322997541961, 299906774892688362286, 658655611067692391840, 1446538998274028918421, 3176885690792900104013, 6977065046637676589991, 15322997781623643899248, 33652282356435747631694, 73906929171669865714372, 162313881771989314409466, 356472526308351006626201, 782882120254535537373895, 1719359175068477322051158, 3776041718757755807143947, 8292909095432841066327168, 18212811043220573594398571, 39998805510180855890238190, 87844996856382377458644750, 192924336519207207211450637, 423698549751517889177307602, 930522591529645092213889600, 2043604500562916062267857702, 4488143835276475740636887635, 9856816516075720448613741962, 21647441215652644553038280647, 47541891683004025230621592287, 104411020674942210093715012401, 229306422636835396187651705393, 503600430230145617118426378807, 1106002127072678450805908619880, 2428990570728880353817750810767, 5334524241272327878859113434559, 11715627504934569958247386669800, 25729741033443156370042205196770, 56507393314397780471788736480925, 124100957166263069054422388400133, 272549247606176724054884198912445, 598569857277276124335468791689884, 1314572968279952867839505883659776, 2887051637243437066723896795822752, 6340513115081517178026623499739632, 13924969683089651853423028411188423, 30581875156221414061060943792273915
Generating function in Maple syntax:
-(4*x^12+10*x^11+8*x^10-2*x^9-17*x^8-11*x^7+10*x^5+6*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{4 x^{12}+10 x^{11}+8 x^{10}-2 x^{9}-17 x^{8}-11 x^{7}+10 x^{5}+6 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:
(-4*x**12 - 10*x**11 - 8*x**10 + 2*x**9 + 17*x**8 + 11*x**7 - 10*x**5 - 6*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)+4*x^12+10*x^11+8*x^10-2*x^9-17*x^8-11*x^7+10*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^{3}+x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+4 x^{12}+10 x^{11}+8 x^{10}-2 x^{9}-17 x^{8}-11 x^{7}+10 x^{5}+6 x^{4}+x^{3}-2 x +1 = 0
Explicit closed form in Maple syntax:
5275423884/1433463493*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))+1665548503/130314863*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))+19629429390/1433463493*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))+4649179830/1433463493*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))-17897568714/1433463493*Sum(_alpha^(3-n),_alpha = RootOf(_Z^9+3*_Z^8+2*_Z^7-_Z^6-4*_Z^5-2*_Z^4-_Z^2+3*_Z-1))-19492604022/1433463493*Sum(_alpha^(2-n),_alpha = RootOf(_Z^9+3*_Z^8+2*_Z^7-_Z^6-4*_Z^5-2*_Z^4-_Z^2+3*_Z-1))-9469436798/1433463493*Sum(_alpha^(1-n),_alpha = RootOf(_Z^9+3*_Z^8+2*_Z^7-_Z^6-4*_Z^5-2*_Z^4-_Z^2+3*_Z-1))-10608399255/1433463493*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))+10412064148/1433463493*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))-2*piecewise(n = 0,-6,n = 1,3,n = 2,-1,n = 3,2,0)
Explicit closed form in latex syntax:
\frac{5275423884 \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{1665548503 \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)}{130314863}+\frac{19629429390 \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)}{1433463493}+\frac{4649179830 \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)}{1433463493}-\frac{17897568714 \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^{3-n}\right)}{1433463493}-\frac{19492604022 \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^{2-n}\right)}{1433463493}-\frac{9469436798 \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^{1-n}\right)}{1433463493}-\frac{10608399255 \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)}{1433463493}+\frac{10412064148 \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}-2 \left(\left\{\begin{array}{cc}-6 & n =0 \\ 3 & 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) = 20
a(5) = 58
a(6) = 145
a(7) = 347
a(8) = 818
a(9) = 1911
a(10) = 4400
a(11) = 10027
a(12) = 22698
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 >= 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) = 20
a \! \left(5\right) = 58
a \! \left(6\right) = 145
a \! \left(7\right) = 347
a \! \left(8\right) = 818
a \! \left(9\right) = 1911
a \! \left(10\right) = 4400
a \! \left(11\right) = 10027
a \! \left(12\right) = 22698
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 13
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15096/
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[28,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[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[4,x]
F[28,x] = F[2,x]+F[29,x]
F[29,x] = F[18,x]+F[30,x]+F[95,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[43,x]
F[32,x] = F[15,x]+F[33,x]
F[33,x] = F[18,x]+F[34,x]+F[39,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[38,x]
F[36,x] = F[37,x]+F[4,x]
F[37,x] = F[22,x]
F[38,x] = F[25,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[15,x]+F[41,x]
F[41,x] = F[18,x]+F[19,x]+F[42,x]
F[42,x] = F[15,x]*F[4,x]
F[43,x] = F[44,x]+F[90,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[53,x]
F[47,x] = F[15,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[4,x]
F[52,x] = F[25,x]
F[53,x] = F[44,x]+F[54,x]
F[54,x] = 2*F[18,x]+F[55,x]+F[67,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[62,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[26,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[25,x]
F[62,x] = F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[58,x]
F[66,x] = F[63,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]+F[86,x]
F[69,x] = F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]+F[75,x]
F[73,x] = F[4,x]+F[74,x]
F[74,x] = F[49,x]
F[75,x] = F[70,x]+F[76,x]
F[76,x] = 2*F[18,x]+F[67,x]+F[77,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[79,x]+F[81,x]
F[79,x] = F[25,x]+F[80,x]
F[80,x] = F[60,x]
F[81,x] = F[82,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[84,x]+F[85,x]
F[84,x] = F[25,x]
F[85,x] = F[82,x]
F[86,x] = F[87,x]
F[87,x] = 2*F[18,x]+F[77,x]+F[88,x]
F[88,x] = F[4,x]*F[89,x]
F[89,x] = F[70,x]
F[90,x] = 2*F[18,x]+F[55,x]+F[91,x]
F[91,x] = F[4,x]*F[92,x]
F[92,x] = F[93,x]+F[94,x]
F[93,x] = F[70,x]
F[94,x] = F[87,x]
F[95,x] = F[4,x]*F[96,x]
F[96,x] = F[109,x]+F[97,x]
F[97,x] = F[2,x]+F[98,x]
F[98,x] = F[105,x]+F[18,x]+F[99,x]
F[99,x] = F[100,x]*F[4,x]
F[100,x] = F[101,x]+F[103,x]
F[101,x] = F[102,x]+F[4,x]
F[102,x] = F[18,x]+F[34,x]+F[42,x]
F[103,x] = F[104,x]+F[70,x]
F[104,x] = 2*F[18,x]+F[77,x]+F[91,x]
F[105,x] = F[106,x]*F[4,x]
F[106,x] = F[107,x]+F[2,x]
F[107,x] = F[108,x]+F[18,x]+F[99,x]
F[108,x] = F[2,x]*F[4,x]
F[109,x] = F[110,x]+F[112,x]
F[110,x] = F[111,x]+F[18,x]+F[30,x]
F[111,x] = F[106,x]*F[4,x]
F[112,x] = F[113,x]+F[118,x]+F[120,x]+F[18,x]
F[113,x] = F[114,x]*F[4,x]
F[114,x] = F[115,x]+F[117,x]
F[115,x] = F[116,x]+F[25,x]
F[116,x] = F[60,x]
F[117,x] = F[82,x]
F[118,x] = F[119,x]*F[4,x]
F[119,x] = F[70,x]
F[120,x] = F[121,x]*F[4,x]
F[121,x] = F[110,x]+F[122,x]
F[122,x] = 2*F[18,x]+F[113,x]+F[88,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_{28}\! \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_{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_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{2}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{95}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{43}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{39}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{22}\! \left(x \right)
F_{38}\! \left(x \right) = F_{25}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{90}\! \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_{53}\! \left(x \right)
F_{47}\! \left(x \right) = F_{15}\! \left(x \right)+F_{48}\! \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_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right)
F_{52}\! \left(x \right) = F_{25}\! \left(x \right)
F_{53}\! \left(x \right) = F_{44}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{67}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{62}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{26}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{25}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{58}\! \left(x \right)
F_{66}\! \left(x \right) = F_{63}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{86}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \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_{75}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{49}\! \left(x \right)
F_{75}\! \left(x \right) = F_{70}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{67}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{81}\! \left(x \right)
F_{79}\! \left(x \right) = F_{25}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{60}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{85}\! \left(x \right)
F_{84}\! \left(x \right) = F_{25}\! \left(x \right)
F_{85}\! \left(x \right) = F_{82}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{77}\! \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_{70}\! \left(x \right)
F_{90}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{4}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)+F_{94}\! \left(x \right)
F_{93}\! \left(x \right) = F_{70}\! \left(x \right)
F_{94}\! \left(x \right) = F_{87}\! \left(x \right)
F_{95}\! \left(x \right) = F_{4}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{109}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{2}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{105}\! \left(x \right)+F_{18}\! \left(x \right)+F_{99}\! \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_{103}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{4}\! \left(x \right)
F_{102}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{42}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{70}\! \left(x \right)
F_{104}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{77}\! \left(x \right)+F_{91}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{4}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{2}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)+F_{18}\! \left(x \right)+F_{99}\! \left(x \right)
F_{108}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)+F_{112}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{18}\! \left(x \right)+F_{30}\! \left(x \right)
F_{111}\! \left(x \right) = F_{106}\! \left(x \right) F_{4}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)+F_{118}\! \left(x \right)+F_{120}\! \left(x \right)+F_{18}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right) F_{4}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{117}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{25}\! \left(x \right)
F_{116}\! \left(x \right) = F_{60}\! \left(x \right)
F_{117}\! \left(x \right) = F_{82}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right) F_{4}\! \left(x \right)
F_{119}\! \left(x \right) = F_{70}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right) F_{4}\! \left(x \right)
F_{121}\! \left(x \right) = F_{110}\! \left(x \right)+F_{122}\! \left(x \right)
F_{122}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{113}\! \left(x \right)+F_{88}\! \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_28(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) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_4(x))
Eq(F_28(x), F_2(x) + F_29(x))
Eq(F_29(x), F_18(x) + F_30(x) + F_95(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_43(x))
Eq(F_32(x), F_15(x) + F_33(x))
Eq(F_33(x), F_18(x) + F_34(x) + F_39(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_38(x))
Eq(F_36(x), F_37(x) + F_4(x))
Eq(F_37(x), F_22(x))
Eq(F_38(x), F_25(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_15(x) + F_41(x))
Eq(F_41(x), F_18(x) + F_19(x) + F_42(x))
Eq(F_42(x), F_15(x)*F_4(x))
Eq(F_43(x), F_44(x) + F_90(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_53(x))
Eq(F_47(x), F_15(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_4(x))
Eq(F_52(x), F_25(x))
Eq(F_53(x), F_44(x) + F_54(x))
Eq(F_54(x), 2*F_18(x) + F_55(x) + F_67(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_62(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_26(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_25(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_58(x))
Eq(F_66(x), F_63(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_86(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_75(x))
Eq(F_73(x), F_4(x) + F_74(x))
Eq(F_74(x), F_49(x))
Eq(F_75(x), F_70(x) + F_76(x))
Eq(F_76(x), 2*F_18(x) + F_67(x) + F_77(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_79(x) + F_81(x))
Eq(F_79(x), F_25(x) + F_80(x))
Eq(F_80(x), F_60(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_84(x) + F_85(x))
Eq(F_84(x), F_25(x))
Eq(F_85(x), F_82(x))
Eq(F_86(x), F_87(x))
Eq(F_87(x), 2*F_18(x) + F_77(x) + F_88(x))
Eq(F_88(x), F_4(x)*F_89(x))
Eq(F_89(x), F_70(x))
Eq(F_90(x), 2*F_18(x) + F_55(x) + F_91(x))
Eq(F_91(x), F_4(x)*F_92(x))
Eq(F_92(x), F_93(x) + F_94(x))
Eq(F_93(x), F_70(x))
Eq(F_94(x), F_87(x))
Eq(F_95(x), F_4(x)*F_96(x))
Eq(F_96(x), F_109(x) + F_97(x))
Eq(F_97(x), F_2(x) + F_98(x))
Eq(F_98(x), F_105(x) + F_18(x) + F_99(x))
Eq(F_99(x), F_100(x)*F_4(x))
Eq(F_100(x), F_101(x) + F_103(x))
Eq(F_101(x), F_102(x) + F_4(x))
Eq(F_102(x), F_18(x) + F_34(x) + F_42(x))
Eq(F_103(x), F_104(x) + F_70(x))
Eq(F_104(x), 2*F_18(x) + F_77(x) + F_91(x))
Eq(F_105(x), F_106(x)*F_4(x))
Eq(F_106(x), F_107(x) + F_2(x))
Eq(F_107(x), F_108(x) + F_18(x) + F_99(x))
Eq(F_108(x), F_2(x)*F_4(x))
Eq(F_109(x), F_110(x) + F_112(x))
Eq(F_110(x), F_111(x) + F_18(x) + F_30(x))
Eq(F_111(x), F_106(x)*F_4(x))
Eq(F_112(x), F_113(x) + F_118(x) + F_120(x) + F_18(x))
Eq(F_113(x), F_114(x)*F_4(x))
Eq(F_114(x), F_115(x) + F_117(x))
Eq(F_115(x), F_116(x) + F_25(x))
Eq(F_116(x), F_60(x))
Eq(F_117(x), F_82(x))
Eq(F_118(x), F_119(x)*F_4(x))
Eq(F_119(x), F_70(x))
Eq(F_120(x), F_121(x)*F_4(x))
Eq(F_121(x), F_110(x) + F_122(x))
Eq(F_122(x), F_113(x) + 2*F_18(x) + F_88(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": [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, 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, 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": [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, 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", 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