0123_0213_0231_0321_3102
Counting sequence:
1, 1, 2, 6, 19, 53, 129, 304, 699, 1570, 3469, 7591, 16487, 35600, 76549, 164100, 350969, 749317, 1597689, 3403208, 7243647, 15409118, 32764997, 69646523, 148006195, 314468800, 668055041, 1419054144, 3014041057, 6401349641, 13594797717, 28870733176, 61309896363, 130194903754, 276471425485, 587085131407, 1246659263807, 2647227703472, 5621243837533, 11936354737692, 25346010672521, 53820341061229, 114283219615953, 242670994798376, 515291291914359, 1094176485400406, 2323387657509269, 4933507245034307, 10475860437238123, 22244544476855968, 47234273208010457, 100297680047004696, 212972968569625969, 452228616470379569, 960266029520474541, 2039036813678570200, 4329707421703949283, 9193735806243575938, 19522052601649856413, 41453282844842061271, 88022231555874799799, 186907106836175147408, 396880034558193035893, 842738211088632580404, 1789476988596915439001, 3799789589533104989269, 8068503242648482192425, 17132723511836314355336, 36379760364253279070703, 77249070274244840590670, 164031285334768227224549, 348305325282284841936203, 739594275002442410220067, 1570460316532239076792192, 3334727820906666416075825, 7080987351175724476928112, 15035824376456997137518081, 31927188036983140350419417, 67794442813382770294672197, 143955254395881812644948024, 305675722182173167707186075, 649074238515065570204972218, 1378249355498074982145744493, 2926585547787836359971985567, 6214334825681281743213393071, 13195567563114877980075159152, 28019572197988354104983705869, 59496980512281318761077662924, 126336357494722214479746207401, 268263617540310070047697375165, 569633080473531925063429829121, 1209563374056909429033394228712, 2568396404647526585396194015719, 5453753174808201034336944729542, 11580542488639712019856371730229, 24590215221055967804667179894483, 52215056868929419477534100930395, 110873863417402680525746872505440, 235430435706567211201031307261449, 499914843305305884975042930753864, 1061523119586746266415201604800401
Generating function in Maple syntax:
(3*x^7+2*x^6-8*x^5-5*x^4-2*x^3+2*x-1)/(x-1)/(x^2+x-1)/(2*x^4+2*x^3+x^2+x-1)
Generating function in latex syntax:
\frac{3 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}-2 x^{3}+2 x -1}{\left(x -1\right) \left(x^{2}+x -1\right) \left(2 x^{4}+2 x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(3*x**7 + 2*x**6 - 8*x**5 - 5*x**4 - 2*x**3 + 2*x - 1)/((x - 1)*(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)*(x^2+x-1)*(2*x^4+2*x^3+x^2+x-1)*F(x)-3*x^7-2*x^6+8*x^5+5*x^4+2*x^3-2*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}+x -1\right) \left(2 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)-3 x^{7}-2 x^{6}+8 x^{5}+5 x^{4}+2 x^{3}-2 x +1 = 0
Explicit closed form in Maple syntax:
13973/1325*Sum(_alpha^(-n+5),_alpha = RootOf(2*_Z^7+2*_Z^6-3*_Z^5-_Z^4-_Z^3-_Z^2+3*_Z-1))+67598/3975*Sum(_alpha^(-n+4),_alpha = RootOf(2*_Z^7+2*_Z^6-3*_Z^5-_Z^4-_Z^3-_Z^2+3*_Z-1))-43891/7950*Sum(_alpha^(-n+3),_alpha = RootOf(2*_Z^7+2*_Z^6-3*_Z^5-_Z^4-_Z^3-_Z^2+3*_Z-1))-4347/530*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^7+2*_Z^6-3*_Z^5-_Z^4-_Z^3-_Z^2+3*_Z-1))-12899/1325*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^7+2*_Z^6-3*_Z^5-_Z^4-_Z^3-_Z^2+3*_Z-1))-45544/3975*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^7+2*_Z^6-3*_Z^5-_Z^4-_Z^3-_Z^2+3*_Z-1))+36427/3975*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^7+2*_Z^6-3*_Z^5-_Z^4-_Z^3-_Z^2+3*_Z-1))+piecewise(n = 0,3/2,0)
Explicit closed form in latex syntax:
\frac{13973 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +5}\right)}{1325}+\frac{67598 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{3975}-\frac{43891 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{7950}-\frac{4347 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{530}-\frac{12899 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{1325}-\frac{45544 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{3975}+\frac{36427 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{3975}+\left(\left\{\begin{array}{cc}\frac{3}{2} & 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) = 53
a(6) = 129
a(7) = 304
a(n+2) = -2*a(n)-4*a(n+1)+a(n+4)+2*a(n+5)-a(n+6)+9, n >= 8
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) = 53
a \! \left(6\right) = 129
a \! \left(7\right) = 304
a \! \left(n +2\right) = -2 a \! \left(n \right)-4 a \! \left(n +1\right)+a \! \left(n +4\right)+2 a \! \left(n +5\right)-a \! \left(n +6\right)+9, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16699/
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[17,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[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]+F[34,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[6,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[31,x]
F[25,x] = F[18,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[18,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[18,x]*F[4,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[28,x]*F[4,x]
F[34,x] = F[35,x]+F[36,x]+F[99,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[51,x]
F[38,x] = F[39,x]+F[7,x]
F[39,x] = F[35,x]+F[40,x]+F[46,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[11,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[4,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[14,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[4,x]
F[51,x] = F[34,x]+F[52,x]
F[52,x] = 2*F[35,x]+F[53,x]+F[95,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[56,x]+F[92,x]
F[56,x] = F[35,x]+F[57,x]+F[71,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[64,x]
F[59,x] = F[11,x]+F[60,x]
F[60,x] = F[35,x]+F[40,x]+F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[15,x]+F[63,x]
F[63,x] = F[49,x]
F[64,x] = F[56,x]+F[65,x]
F[65,x] = 2*F[35,x]+F[53,x]+F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[32,x]+F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[29,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[18,x]+F[73,x]
F[73,x] = F[30,x]+F[35,x]+F[74,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[76,x]+F[84,x]
F[76,x] = F[4,x]+F[77,x]
F[77,x] = F[35,x]+F[78,x]+F[83,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[80,x]
F[80,x] = F[4,x]+F[81,x]
F[81,x] = F[82,x]
F[82,x] = x^2
F[83,x] = F[15,x]*F[4,x]
F[84,x] = F[29,x]+F[85,x]
F[85,x] = 2*F[35,x]+F[86,x]+F[91,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[88,x]
F[88,x] = F[29,x]+F[89,x]
F[89,x] = F[90,x]
F[90,x] = F[29,x]*F[4,x]
F[91,x] = F[32,x]*F[4,x]
F[92,x] = F[93,x]
F[93,x] = F[4,x]*F[94,x]
F[94,x] = F[29,x]
F[95,x] = F[4,x]*F[96,x]
F[96,x] = F[97,x]+F[98,x]
F[97,x] = F[32,x]
F[98,x] = F[69,x]
F[99,x] = F[100,x]*F[4,x]
F[100,x] = F[101,x]+F[102,x]
F[101,x] = F[18,x]+F[56,x]
F[102,x] = F[103,x]
F[103,x] = F[104,x]+F[33,x]+F[35,x]
F[104,x] = F[105,x]*F[4,x]
F[105,x] = F[106,x]+F[110,x]
F[106,x] = F[107,x]+F[15,x]
F[107,x] = F[108,x]+F[35,x]+F[78,x]
F[108,x] = F[109,x]*F[4,x]
F[109,x] = F[15,x]
F[110,x] = F[111,x]+F[32,x]
F[111,x] = 2*F[35,x]+F[112,x]+F[86,x]
F[112,x] = F[113,x]*F[4,x]
F[113,x] = F[32,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_{17}\! \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_{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_{34}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
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_{6}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
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_{31}\! \left(x \right)
F_{25}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \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_{18}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{99}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{51}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{7}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{46}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{11}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{14}\! \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_{4}\! \left(x \right)
F_{51}\! \left(x \right) = F_{34}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{53}\! \left(x \right)+F_{95}\! \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_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{92}\! \left(x \right)
F_{56}\! \left(x \right) = F_{35}\! \left(x \right)+F_{57}\! \left(x \right)+F_{71}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{64}\! \left(x \right)
F_{59}\! \left(x \right) = F_{11}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{15}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{49}\! \left(x \right)
F_{64}\! \left(x \right) = F_{56}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{53}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{32}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{29}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{18}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{84}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{35}\! \left(x \right)+F_{78}\! \left(x \right)+F_{83}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \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_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = x^{2}
F_{83}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{84}\! \left(x \right) = F_{29}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{86}\! \left(x \right)+F_{91}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{29}\! \left(x \right)+F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{91}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{4}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{29}\! \left(x \right)
F_{95}\! \left(x \right) = F_{4}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)+F_{98}\! \left(x \right)
F_{97}\! \left(x \right) = F_{32}\! \left(x \right)
F_{98}\! \left(x \right) = F_{69}\! \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_{102}\! \left(x \right)
F_{101}\! \left(x \right) = F_{18}\! \left(x \right)+F_{56}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right) F_{4}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{110}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{15}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)+F_{35}\! \left(x \right)+F_{78}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{4}\! \left(x \right)
F_{109}\! \left(x \right) = F_{15}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{32}\! \left(x \right)
F_{111}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{112}\! \left(x \right)+F_{86}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{4}\! \left(x \right)
F_{113}\! \left(x \right) = F_{32}\! \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_17(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))
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_34(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_6(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_31(x))
Eq(F_25(x), F_18(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_18(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_18(x)*F_4(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_28(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_99(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_51(x))
Eq(F_38(x), F_39(x) + F_7(x))
Eq(F_39(x), F_35(x) + F_40(x) + F_46(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_11(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_4(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_14(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_4(x))
Eq(F_51(x), F_34(x) + F_52(x))
Eq(F_52(x), 2*F_35(x) + F_53(x) + F_95(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_56(x) + F_92(x))
Eq(F_56(x), F_35(x) + F_57(x) + F_71(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_64(x))
Eq(F_59(x), F_11(x) + F_60(x))
Eq(F_60(x), F_35(x) + F_40(x) + F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_15(x) + F_63(x))
Eq(F_63(x), F_49(x))
Eq(F_64(x), F_56(x) + F_65(x))
Eq(F_65(x), 2*F_35(x) + F_53(x) + F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_32(x) + F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_29(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_18(x) + F_73(x))
Eq(F_73(x), F_30(x) + F_35(x) + F_74(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_84(x))
Eq(F_76(x), F_4(x) + F_77(x))
Eq(F_77(x), F_35(x) + F_78(x) + F_83(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_4(x) + F_81(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), x**2)
Eq(F_83(x), F_15(x)*F_4(x))
Eq(F_84(x), F_29(x) + F_85(x))
Eq(F_85(x), 2*F_35(x) + F_86(x) + F_91(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_88(x))
Eq(F_88(x), F_29(x) + F_89(x))
Eq(F_89(x), F_90(x))
Eq(F_90(x), F_29(x)*F_4(x))
Eq(F_91(x), F_32(x)*F_4(x))
Eq(F_92(x), F_93(x))
Eq(F_93(x), F_4(x)*F_94(x))
Eq(F_94(x), F_29(x))
Eq(F_95(x), F_4(x)*F_96(x))
Eq(F_96(x), F_97(x) + F_98(x))
Eq(F_97(x), F_32(x))
Eq(F_98(x), F_69(x))
Eq(F_99(x), F_100(x)*F_4(x))
Eq(F_100(x), F_101(x) + F_102(x))
Eq(F_101(x), F_18(x) + F_56(x))
Eq(F_102(x), F_103(x))
Eq(F_103(x), F_104(x) + F_33(x) + F_35(x))
Eq(F_104(x), F_105(x)*F_4(x))
Eq(F_105(x), F_106(x) + F_110(x))
Eq(F_106(x), F_107(x) + F_15(x))
Eq(F_107(x), F_108(x) + F_35(x) + F_78(x))
Eq(F_108(x), F_109(x)*F_4(x))
Eq(F_109(x), F_15(x))
Eq(F_110(x), F_111(x) + F_32(x))
Eq(F_111(x), F_112(x) + 2*F_35(x) + F_86(x))
Eq(F_112(x), F_113(x)*F_4(x))
Eq(F_113(x), F_32(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, 1, 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": [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, 2, 1, 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": [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, 2, 1, 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": [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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [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, 0], [0, 1], [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": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [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": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"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, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 2], [1, 0], [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, 0], [1, 2], [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": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 0], [1, 0], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 2], [1, 0], [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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": 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