0132_0321_2130
Counting sequence:
1, 1, 2, 6, 21, 73, 242, 772, 2409, 7439, 22872, 70204, 215345, 660375, 2024866, 6208416, 19035179, 58361985, 178937794, 548622648, 1682074579, 5157232471, 15812048862, 48479661510, 148638395025, 455724560391, 1397249174878, 4283958833718, 13134595902579, 40270603946867, 123469466000352, 378556752077848, 1160653068209771, 3558556378535267, 10910515679542896, 33451585342745574, 102562389790712591, 314455763211328697, 964119763771076614, 2955986271014251150, 9063038807800519239, 27787230690863466641, 85195507361470784896, 261208990393021324546, 800865430293783572433, 2455449318473315827691, 7528395068044518261968, 23082020823706507047242, 70769357942902032790473, 216978489964223103753905, 665254941907755769253628, 2039668253777895231676904, 6253612447491187665053111, 19173543820658620649119185, 58785987415993357008403436, 180237537139580343041083048, 552607368219582828351917257, 1694291368251907655191509069, 5194688680647951336412983294, 15926888960482414552914100218, 48831760198552650002903365361, 149717927336935777635711891693, 459034400458338687352905272286, 1407397127065110936593326586808, 4315072402620290327587567268773, 13229989945115063401390136773971, 40563081593152093567697621284772, 124366200969051771047293329941968, 381306136910606969849441502314061, 1169082668062455209039963947482235, 3584401488624478496599945785925198, 10989756655058896057531521206084948, 33694537768914624838438904370773095, 103307280688348394790062044811700293, 316739594898588290652319745963063502, 971121980058438876695111659504959564, 2977455030384097572110556834760648367, 9128861914366398091978005029881612027, 27989043999371105438566712595931465994, 85814265934496147708393197264919056122, 263106101017308486533057207096793543117, 806681961777438192884151435476372307595, 2473282774291453895237918476270083657066, 7583072352489931043430994952687156517386, 23249661098525474432374572587416132400655, 71283342169193224695322029778521250851863, 218554362976608937531154695909547123365132, 670086560514197349032957259526104223784140, 2054481971745416868918053793628960623165031, 6299031231111381665654049703745756687407975, 19312797579239736945094209594560733771482172, 59212938728497154784599112379188131272737914, 181546567682856716283984125478417161757361963, 556620849178761738427479096113014381630469853, 1706596680371955571467527844202930065459658802, 5232416704752686654336497105701079144010597378, 16042562889673407436366648196716221544274285891, 49186415874591709470559000855166956264243326805, 150805299828096183299282242753606702844177215500, 462368278961955719477832750092309256628838274198, 1417618781527804282416085518025428911151928367597
Generating function in Maple syntax:
-(4*x^6-16*x^5+30*x^4-31*x^3+20*x^2-7*x+1)/(2*x^4-7*x^3+8*x^2-5*x+1)/(x-1)^3
Generating function in latex syntax:
-\frac{4 x^{6}-16 x^{5}+30 x^{4}-31 x^{3}+20 x^{2}-7 x +1}{\left(2 x^{4}-7 x^{3}+8 x^{2}-5 x +1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-4*x**6 + 16*x**5 - 30*x**4 + 31*x**3 - 20*x**2 + 7*x - 1)/((x - 1)**3*(2*x**4 - 7*x**3 + 8*x**2 - 5*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x^4-7*x^3+8*x^2-5*x+1)*(x-1)^3*F(x)+4*x^6-16*x^5+30*x^4-31*x^3+20*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{4}-7 x^{3}+8 x^{2}-5 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+4 x^{6}-16 x^{5}+30 x^{4}-31 x^{3}+20 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
-1/2*n^2+1/2*n-1+384/751*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))-48/751*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))-128/751*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))+110/751*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))
Explicit closed form in latex syntax:
-\frac{n^{2}}{2}+\frac{n}{2}-1+\frac{384 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{751}-\frac{48 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{751}-\frac{128 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{751}+\frac{110 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{751}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 73
a(6) = 242
a(n+4) = 1/2*n^2-2*a(n)+7*a(n+1)-8*a(n+2)+5*a(n+3)+3/2*n+2, n >= 7
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) = 21
a \! \left(5\right) = 73
a \! \left(6\right) = 242
a \! \left(n +4\right) = \frac{n^{2}}{2}-2 a \! \left(n \right)+7 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right)+\frac{3 n}{2}+2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14702/
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[12,x]*F[4,x]
F[4,x] = F[20,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[16,x]
F[19,x] = F[12,x]*F[13,x]
F[20,x] = F[21,x]+F[41,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[31,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[9,x]
F[28,x] = F[10,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[28,x]
F[31,x] = F[21,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[21,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[35,x]
F[38,x] = F[36,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[38,x]
F[41,x] = F[15,x]+F[42,x]+F[93,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[59,x]
F[44,x] = F[45,x]+F[6,x]
F[45,x] = F[15,x]+F[46,x]+F[51,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[17,x]+F[48,x]
F[48,x] = F[14,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[48,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[13,x]+F[53,x]
F[53,x] = F[18,x]+F[54,x]
F[54,x] = 2*F[15,x]+F[55,x]+F[58,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[18,x]+F[57,x]
F[57,x] = F[55,x]
F[58,x] = F[12,x]*F[53,x]
F[59,x] = F[41,x]+F[60,x]
F[60,x] = 2*F[15,x]+F[61,x]+F[81,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[63,x]+F[78,x]
F[63,x] = F[64,x]+F[77,x]
F[64,x] = F[15,x]+F[65,x]+F[75,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[67,x]+F[69,x]
F[67,x] = F[10,x]+F[68,x]
F[68,x] = F[15,x]+F[19,x]+F[46,x]
F[69,x] = F[64,x]+F[70,x]
F[70,x] = 2*F[15,x]+F[61,x]+F[71,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[36,x]+F[73,x]
F[73,x] = 2*F[15,x]+F[71,x]+F[74,x]
F[74,x] = F[12,x]*F[63,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[21,x]+F[64,x]
F[77,x] = F[74,x]
F[78,x] = F[73,x]+F[79,x]
F[79,x] = F[80,x]
F[80,x] = F[12,x]*F[78,x]
F[81,x] = F[12,x]*F[82,x]
F[82,x] = F[72,x]+F[83,x]
F[83,x] = F[84,x]+F[88,x]
F[84,x] = F[85,x]
F[85,x] = F[12,x]*F[86,x]
F[86,x] = F[36,x]+F[87,x]
F[87,x] = F[85,x]
F[88,x] = 3*F[15,x]+F[89,x]+F[92,x]
F[89,x] = F[12,x]*F[90,x]
F[90,x] = F[77,x]+F[91,x]
F[91,x] = F[89,x]
F[92,x] = F[12,x]*F[83,x]
F[93,x] = F[12,x]*F[94,x]
F[94,x] = F[76,x]+F[95,x]
F[95,x] = F[113,x]+F[96,x]
F[96,x] = F[15,x]+F[37,x]+F[97,x]
F[97,x] = F[12,x]*F[98,x]
F[98,x] = F[106,x]+F[99,x]
F[99,x] = F[10,x]+F[100,x]
F[100,x] = F[101,x]+F[15,x]+F[30,x]
F[101,x] = F[102,x]*F[12,x]
F[102,x] = F[103,x]+F[17,x]
F[103,x] = F[104,x]+F[29,x]
F[104,x] = F[105,x]
F[105,x] = F[103,x]*F[12,x]
F[106,x] = F[107,x]+F[36,x]
F[107,x] = 2*F[15,x]+F[108,x]+F[40,x]
F[108,x] = F[109,x]*F[12,x]
F[109,x] = F[110,x]+F[86,x]
F[110,x] = F[111,x]+F[39,x]
F[111,x] = F[112,x]
F[112,x] = F[110,x]*F[12,x]
F[113,x] = F[114,x]+F[133,x]+F[15,x]+F[74,x]
F[114,x] = F[115,x]*F[12,x]
F[115,x] = F[116,x]+F[123,x]
F[116,x] = F[117,x]+F[14,x]
F[117,x] = F[118,x]+F[15,x]+F[50,x]+F[58,x]
F[118,x] = F[119,x]*F[12,x]
F[119,x] = F[120,x]+F[56,x]
F[120,x] = F[121,x]+F[49,x]
F[121,x] = F[122,x]
F[122,x] = F[12,x]*F[120,x]
F[123,x] = F[124,x]+F[73,x]
F[124,x] = 2*F[15,x]+F[125,x]+F[130,x]+F[80,x]
F[125,x] = F[12,x]*F[126,x]
F[126,x] = F[127,x]+F[90,x]
F[127,x] = F[128,x]+F[79,x]
F[128,x] = F[129,x]
F[129,x] = F[12,x]*F[127,x]
F[130,x] = F[12,x]*F[131,x]
F[131,x] = F[132,x]+F[87,x]
F[132,x] = 3*F[15,x]+F[130,x]+F[89,x]
F[133,x] = F[12,x]*F[95,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{20}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{41}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{9}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{21}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{21}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{93}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{59}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{6}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{51}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{17}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{13}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{18}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{58}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{18}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{59}\! \left(x \right) = F_{41}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{61}\! \left(x \right)+F_{81}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{78}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{77}\! \left(x \right)
F_{64}\! \left(x \right) = F_{15}\! \left(x \right)+F_{65}\! \left(x \right)+F_{75}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{69}\! \left(x \right)
F_{67}\! \left(x \right) = F_{10}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{46}\! \left(x \right)
F_{69}\! \left(x \right) = F_{64}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{61}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{36}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{71}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{21}\! \left(x \right)+F_{64}\! \left(x \right)
F_{77}\! \left(x \right) = F_{74}\! \left(x \right)
F_{78}\! \left(x \right) = F_{73}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{72}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{88}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{12}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{36}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{85}\! \left(x \right)
F_{88}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{89}\! \left(x \right)+F_{92}\! \left(x \right)
F_{89}\! \left(x \right) = F_{12}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{77}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{89}\! \left(x \right)
F_{92}\! \left(x \right) = F_{12}\! \left(x \right) F_{83}\! \left(x \right)
F_{93}\! \left(x \right) = F_{12}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{76}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{113}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{12}\! \left(x \right) F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{106}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{10}\! \left(x \right)+F_{100}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{15}\! \left(x \right)+F_{30}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{12}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{17}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{29}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)
F_{105}\! \left(x \right) = F_{103}\! \left(x \right) F_{12}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{36}\! \left(x \right)
F_{107}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{108}\! \left(x \right)+F_{40}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{12}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)+F_{86}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{39}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{110}\! \left(x \right) F_{12}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)+F_{133}\! \left(x \right)+F_{15}\! \left(x \right)+F_{74}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right) F_{12}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{123}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)+F_{14}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)+F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{58}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right) F_{12}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)+F_{56}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)+F_{49}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)
F_{122}\! \left(x \right) = F_{12}\! \left(x \right) F_{120}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)+F_{73}\! \left(x \right)
F_{124}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{125}\! \left(x \right)+F_{130}\! \left(x \right)+F_{80}\! \left(x \right)
F_{125}\! \left(x \right) = F_{12}\! \left(x \right) F_{126}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)+F_{90}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)+F_{79}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right)
F_{129}\! \left(x \right) = F_{12}\! \left(x \right) F_{127}\! \left(x \right)
F_{130}\! \left(x \right) = F_{12}\! \left(x \right) F_{131}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)+F_{87}\! \left(x \right)
F_{132}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{130}\! \left(x \right)+F_{89}\! \left(x \right)
F_{133}\! \left(x \right) = F_{12}\! \left(x \right) F_{95}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_20(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_12(x)*F_13(x))
Eq(F_20(x), F_21(x) + F_41(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_31(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_9(x))
Eq(F_28(x), F_10(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_28(x))
Eq(F_31(x), F_21(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_21(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_35(x))
Eq(F_38(x), F_36(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_38(x))
Eq(F_41(x), F_15(x) + F_42(x) + F_93(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_59(x))
Eq(F_44(x), F_45(x) + F_6(x))
Eq(F_45(x), F_15(x) + F_46(x) + F_51(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_17(x) + F_48(x))
Eq(F_48(x), F_14(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_48(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_13(x) + F_53(x))
Eq(F_53(x), F_18(x) + F_54(x))
Eq(F_54(x), 2*F_15(x) + F_55(x) + F_58(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_18(x) + F_57(x))
Eq(F_57(x), F_55(x))
Eq(F_58(x), F_12(x)*F_53(x))
Eq(F_59(x), F_41(x) + F_60(x))
Eq(F_60(x), 2*F_15(x) + F_61(x) + F_81(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_78(x))
Eq(F_63(x), F_64(x) + F_77(x))
Eq(F_64(x), F_15(x) + F_65(x) + F_75(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_69(x))
Eq(F_67(x), F_10(x) + F_68(x))
Eq(F_68(x), F_15(x) + F_19(x) + F_46(x))
Eq(F_69(x), F_64(x) + F_70(x))
Eq(F_70(x), 2*F_15(x) + F_61(x) + F_71(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_36(x) + F_73(x))
Eq(F_73(x), 2*F_15(x) + F_71(x) + F_74(x))
Eq(F_74(x), F_12(x)*F_63(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_21(x) + F_64(x))
Eq(F_77(x), F_74(x))
Eq(F_78(x), F_73(x) + F_79(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_12(x)*F_78(x))
Eq(F_81(x), F_12(x)*F_82(x))
Eq(F_82(x), F_72(x) + F_83(x))
Eq(F_83(x), F_84(x) + F_88(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), F_12(x)*F_86(x))
Eq(F_86(x), F_36(x) + F_87(x))
Eq(F_87(x), F_85(x))
Eq(F_88(x), 3*F_15(x) + F_89(x) + F_92(x))
Eq(F_89(x), F_12(x)*F_90(x))
Eq(F_90(x), F_77(x) + F_91(x))
Eq(F_91(x), F_89(x))
Eq(F_92(x), F_12(x)*F_83(x))
Eq(F_93(x), F_12(x)*F_94(x))
Eq(F_94(x), F_76(x) + F_95(x))
Eq(F_95(x), F_113(x) + F_96(x))
Eq(F_96(x), F_15(x) + F_37(x) + F_97(x))
Eq(F_97(x), F_12(x)*F_98(x))
Eq(F_98(x), F_106(x) + F_99(x))
Eq(F_99(x), F_10(x) + F_100(x))
Eq(F_100(x), F_101(x) + F_15(x) + F_30(x))
Eq(F_101(x), F_102(x)*F_12(x))
Eq(F_102(x), F_103(x) + F_17(x))
Eq(F_103(x), F_104(x) + F_29(x))
Eq(F_104(x), F_105(x))
Eq(F_105(x), F_103(x)*F_12(x))
Eq(F_106(x), F_107(x) + F_36(x))
Eq(F_107(x), F_108(x) + 2*F_15(x) + F_40(x))
Eq(F_108(x), F_109(x)*F_12(x))
Eq(F_109(x), F_110(x) + F_86(x))
Eq(F_110(x), F_111(x) + F_39(x))
Eq(F_111(x), F_112(x))
Eq(F_112(x), F_110(x)*F_12(x))
Eq(F_113(x), F_114(x) + F_133(x) + F_15(x) + F_74(x))
Eq(F_114(x), F_115(x)*F_12(x))
Eq(F_115(x), F_116(x) + F_123(x))
Eq(F_116(x), F_117(x) + F_14(x))
Eq(F_117(x), F_118(x) + F_15(x) + F_50(x) + F_58(x))
Eq(F_118(x), F_119(x)*F_12(x))
Eq(F_119(x), F_120(x) + F_56(x))
Eq(F_120(x), F_121(x) + F_49(x))
Eq(F_121(x), F_122(x))
Eq(F_122(x), F_12(x)*F_120(x))
Eq(F_123(x), F_124(x) + F_73(x))
Eq(F_124(x), F_125(x) + F_130(x) + 2*F_15(x) + F_80(x))
Eq(F_125(x), F_12(x)*F_126(x))
Eq(F_126(x), F_127(x) + F_90(x))
Eq(F_127(x), F_128(x) + F_79(x))
Eq(F_128(x), F_129(x))
Eq(F_129(x), F_12(x)*F_127(x))
Eq(F_130(x), F_12(x)*F_131(x))
Eq(F_131(x), F_132(x) + F_87(x))
Eq(F_132(x), F_130(x) + 3*F_15(x) + F_89(x))
Eq(F_133(x), F_12(x)*F_95(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, 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": [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, 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": [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, 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": [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, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"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, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "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": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 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