0123_1032_1302
Counting sequence:
1, 1, 2, 6, 21, 73, 247, 821, 2704, 8868, 29030, 94960, 310531, 1015359, 3319829, 10854379, 35488838, 116031978, 379370276, 1240362982, 4055405209, 13259272613, 43351600979, 141739396705, 463421329340, 1515170329456, 4953896123490, 16196916164572, 52956316947055, 173142311541835, 566094127655697, 1850862209891799, 6051451079684210, 19785405944373302, 64688994958224672, 211502664159147730, 691514483650664069, 2260927932041663025, 7392173605532955983, 24168939593308163661, 79021093420717113448, 258361902114009875516, 844722207380504489822, 2761845310021422560456, 9029938422172196524251, 29523662173385527396183, 96528524047068365133837, 315602986512461271396291, 1031873698255411044977054, 3373742881578397095988098, 11030556404572277791814940, 36064744370064741969250750, 117914794029698928693535601, 385526055817746044450021309, 1260489329922016875657981259, 4121208740294227226668933689, 13474419083046350368601835028, 44055028771146753545716503048, 144039275316036417530398376026, 470940853116798110221647666612, 1539755644061376695386828546119, 5034278567527185285943167938403, 16459729044158207278055481450249, 53815591762174026645166079079855, 175951737051275855318384207847370, 575279631006897755894149557052366, 1880894496398111510911777762829080, 6149642566674304486601430795856042, 20106446039516678925458689009043805, 65738645450840025263854019174267465, 214934528818158673252242871261652807, 702735070999147975068326805541898789, 2297613988443795931044286646731174080, 7512119798414057145476651912822308052, 24561107370323136738910401322179510102, 80303298062937967381300222843713467040, 262554109737610573599422991233959956659, 858428759502274665731612099049058448495, 2806659304922142876890779868181780860933, 9176459160656968657567129158047868014491, 30002716246865995331059401225744246375158, 98094806115337361105317777165196943182106, 320723994042065395743227163299650769635092, 1048616990315981490319074124154500741747606, 3428485591368404496257250561049444544452681, 11209539382609804305558234198333577491841493, 36649934736965966714099333358494977817506627, 119828091982771262932237131961078067883410961, 391781642485406760197766996165494493064810604, 1280942163467245185422555255495733724244026528, 4188079910378302986961394297408607553548500050, 13693056436082289911973445415478218027591545740, 44769870340128741057854709280678516898208739871, 146376471873024700469421289471096628617157326459, 478582389343877158528067920873307399829186181889, 1564739882436692014819717233641296747261690232199, 5115965305461185984633507750887458046793517040418, 16726806353222422211131015144372556745047369833318, 54688809261379514920262971792110446334332308680976, 178806748596773165016925857237099031195774159309442, 584614179309398959729176543860809899270194450866229
Generating function in Maple syntax:
(x^3-2*x^2+3*x-1)^2/(2*x^3-3*x^2+4*x-1)/(x-1)^3
Generating function in latex syntax:
\frac{\left(x^{3}-2 x^{2}+3 x -1\right)^{2}}{\left(2 x^{3}-3 x^{2}+4 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**3 - 2*x**2 + 3*x - 1)**2/((x - 1)**3*(2*x**3 - 3*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x^3-3*x^2+4*x-1)*(x-1)^3*F(x)-(x^3-2*x^2+3*x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{3}-3 x^{2}+4 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)-\left(x^{3}-2 x^{2}+3 x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/68400*(((165*I*3^(1/3)-55*3^(5/6))*38^(1/2)-285*I*3^(5/6)+285*3^(1/3))*(9+2*38^(1/2)*3^(1/2))^(1/3)+5700+((-3*I*3^(2/3)-3*3^(1/6))*38^(1/2)-399*I*3^(1/6)-133*3^(2/3))*(9+2*38^(1/2)*3^(1/2))^(2/3))*(1/12*(27+6*38^(1/2)*3^(1/2))^(1/3)-1/12*I*3^(5/6)*(9+2*38^(1/2)*3^(1/2))^(1/3)+1/2+1/300*((9*I-2*38^(1/2))*3^(1/6)+(-2*I*38^(1/2)+3)*3^(2/3))*(9+2*38^(1/2)*3^(1/2))^(2/3))^(-n)+1/68400*(((-165*I*3^(1/3)-55*3^(5/6))*38^(1/2)+285*I*3^(5/6)+285*3^(1/3))*(9+2*38^(1/2)*3^(1/2))^(1/3)+5700+((3*I*3^(2/3)-3*3^(1/6))*38^(1/2)+399*I*3^(1/6)-133*3^(2/3))*(9+2*38^(1/2)*3^(1/2))^(2/3))*(1/12*(27+6*38^(1/2)*3^(1/2))^(1/3)+1/12*I*3^(5/6)*(9+2*38^(1/2)*3^(1/2))^(1/3)+1/2+1/300*((-9*I-2*38^(1/2))*3^(1/6)+(2*I*38^(1/2)+3)*3^(2/3))*(9+2*38^(1/2)*3^(1/2))^(2/3))^(-n)+1/68400*((110*3^(5/6)*38^(1/2)-570*3^(1/3))*(9+2*38^(1/2)*3^(1/2))^(1/3)+5700+(6*38^(1/2)*3^(1/6)+266*3^(2/3))*(9+2*38^(1/2)*3^(1/2))^(2/3))*(1/150*(2*38^(1/2)*3^(1/6)-3*3^(2/3))*(9+2*38^(1/2)*3^(1/2))^(2/3)-1/6*(27+6*38^(1/2)*3^(1/2))^(1/3)+1/2)^(-n)-1/4*n^2+1/4*n+1/4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(165 \,\mathrm{I} \,3^{\frac{1}{3}}-55 \,3^{\frac{5}{6}}\right) \sqrt{38}-285 \,\mathrm{I} \,3^{\frac{5}{6}}+285 \,3^{\frac{1}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}+5700+\left(\left(-3 \,\mathrm{I} \,3^{\frac{2}{3}}-3 \,3^{\frac{1}{6}}\right) \sqrt{38}-399 \,\mathrm{I} \,3^{\frac{1}{6}}-133 \,3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(27+6 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \,3^{\frac{5}{6}} \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{2}+\frac{\left(\left(9 \,\mathrm{I}-2 \sqrt{38}\right) 3^{\frac{1}{6}}+\left(-2 \,\mathrm{I} \sqrt{38}+3\right) 3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}\right)^{-n}}{68400}\\+\\\frac{\left(\left(\left(-165 \,\mathrm{I} \,3^{\frac{1}{3}}-55 \,3^{\frac{5}{6}}\right) \sqrt{38}+285 \,\mathrm{I} \,3^{\frac{5}{6}}+285 \,3^{\frac{1}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}+5700+\left(\left(3 \,\mathrm{I} \,3^{\frac{2}{3}}-3 \,3^{\frac{1}{6}}\right) \sqrt{38}+399 \,\mathrm{I} \,3^{\frac{1}{6}}-133 \,3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(27+6 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \,3^{\frac{5}{6}} \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{2}+\frac{\left(\left(-9 \,\mathrm{I}-2 \sqrt{38}\right) 3^{\frac{1}{6}}+\left(2 \,\mathrm{I} \sqrt{38}+3\right) 3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}\right)^{-n}}{68400}\\+\\\frac{\left(\left(110 \,3^{\frac{5}{6}} \sqrt{38}-570 \,3^{\frac{1}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}+5700+\left(6 \sqrt{38}\, 3^{\frac{1}{6}}+266 \,3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(2 \sqrt{38}\, 3^{\frac{1}{6}}-3 \,3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}}{150}-\frac{\left(27+6 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{2}\right)^{-n}}{68400}\\-\frac{n^{2}}{4}+\frac{n}{4}+\frac{1}{4} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 73
a(6) = 247
a(n+3) = 2*a(n)-3*a(n+1)+4*a(n+2)+1/2*n*(n+1), 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) = 247
a \! \left(n +3\right) = 2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)+\frac{n \left(n +1\right)}{2}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14660/
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[29,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]+F[16,x]+F[27,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]+F[4,x]
F[19,x] = F[20,x]
F[20,x] = F[11,x]*F[4,x]
F[21,x] = F[22,x]+F[24,x]
F[22,x] = F[15,x]+F[16,x]+F[23,x]
F[23,x] = F[4,x]*F[7,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[26,x]+F[7,x]
F[29,x] = F[2,x]+F[30,x]
F[30,x] = F[113,x]+F[15,x]+F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[40,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[1,x]+F[4,x]
F[37,x] = F[15,x]+F[16,x]+F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[22,x]+F[7,x]
F[40,x] = F[102,x]+F[41,x]
F[41,x] = F[15,x]+F[31,x]+F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[2,x]+F[44,x]
F[44,x] = F[15,x]+F[45,x]+F[90,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[22,x]+F[4,x]
F[48,x] = F[44,x]+F[49,x]
F[49,x] = F[15,x]+F[50,x]+F[94,x]+F[96,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[59,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[4,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[22,x]
F[59,x] = F[60,x]+F[63,x]
F[60,x] = 2*F[15,x]+F[50,x]+F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[44,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]
F[66,x] = 2*F[15,x]+F[67,x]+F[94,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]+F[75,x]
F[69,x] = F[44,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[73,x] = F[4,x]*F[74,x]
F[74,x] = F[2,x]+F[72,x]
F[75,x] = F[66,x]+F[76,x]
F[76,x] = F[77,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[78,x]+F[81,x]
F[81,x] = F[82,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[74,x]+F[84,x]
F[84,x] = F[81,x]+F[85,x]
F[85,x] = 2*F[15,x]+F[79,x]+F[86,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[88,x]+F[92,x]
F[88,x] = F[89,x]+F[91,x]
F[89,x] = F[90,x]
F[90,x] = F[2,x]*F[4,x]
F[91,x] = F[71,x]
F[92,x] = F[93,x]+F[95,x]
F[93,x] = 2*F[15,x]+F[86,x]+F[94,x]
F[94,x] = F[4,x]*F[81,x]
F[95,x] = F[77,x]
F[96,x] = F[4,x]*F[97,x]
F[97,x] = F[100,x]+F[98,x]
F[98,x] = F[44,x]+F[99,x]
F[99,x] = F[71,x]
F[100,x] = F[101,x]+F[66,x]
F[101,x] = F[77,x]
F[102,x] = F[103,x]+F[111,x]+F[15,x]+F[96,x]
F[103,x] = F[104,x]*F[4,x]
F[104,x] = F[105,x]+F[108,x]
F[105,x] = F[106,x]+F[107,x]
F[106,x] = F[54,x]
F[107,x] = F[57,x]
F[108,x] = F[109,x]+F[110,x]
F[109,x] = 2*F[15,x]+F[103,x]+F[61,x]
F[110,x] = F[64,x]
F[111,x] = F[112,x]*F[4,x]
F[112,x] = F[66,x]+F[81,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[2,x]
F[116,x] = F[15,x]+F[45,x]+F[73,x]
F[117,x] = F[118,x]+F[81,x]
F[118,x] = 2*F[15,x]+F[67,x]+F[79,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_{29}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{27}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \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_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{2}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{113}\! \left(x \right)+F_{15}\! \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_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{40}\! \left(x \right) = F_{102}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{2}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{90}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{22}\! \left(x \right)+F_{4}\! \left(x \right)
F_{48}\! \left(x \right) = F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{94}\! \left(x \right)+F_{96}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{59}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{22}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{50}\! \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_{44}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{67}\! \left(x \right)+F_{94}\! \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_{75}\! \left(x \right)
F_{69}\! \left(x \right) = F_{44}\! \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_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{2}\! \left(x \right)+F_{72}\! \left(x \right)
F_{75}\! \left(x \right) = F_{66}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \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_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{78}\! \left(x \right)+F_{81}\! \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_{74}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{81}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{79}\! \left(x \right)+F_{86}\! \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_{92}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{91}\! \left(x \right)
F_{89}\! \left(x \right) = F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{91}\! \left(x \right) = F_{71}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)+F_{95}\! \left(x \right)
F_{93}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{86}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{95}\! \left(x \right) = F_{77}\! \left(x \right)
F_{96}\! \left(x \right) = F_{4}\! \left(x \right) F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{100}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{44}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{71}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{66}\! \left(x \right)
F_{101}\! \left(x \right) = F_{77}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{111}\! \left(x \right)+F_{15}\! \left(x \right)+F_{96}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{4}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{108}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{107}\! \left(x \right)
F_{106}\! \left(x \right) = F_{54}\! \left(x \right)
F_{107}\! \left(x \right) = F_{57}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{110}\! \left(x \right)
F_{109}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{103}\! \left(x \right)+F_{61}\! \left(x \right)
F_{110}\! \left(x \right) = F_{64}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right) F_{4}\! \left(x \right)
F_{112}\! \left(x \right) = F_{66}\! \left(x \right)+F_{81}\! \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_{2}\! \left(x \right)
F_{116}\! \left(x \right) = F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{73}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)+F_{81}\! \left(x \right)
F_{118}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{67}\! \left(x \right)+F_{79}\! \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_29(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_27(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x) + F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_11(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_24(x))
Eq(F_22(x), F_15(x) + F_16(x) + F_23(x))
Eq(F_23(x), F_4(x)*F_7(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_26(x) + F_7(x))
Eq(F_29(x), F_2(x) + F_30(x))
Eq(F_30(x), F_113(x) + F_15(x) + F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_40(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_1(x) + F_4(x))
Eq(F_37(x), F_15(x) + F_16(x) + F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_22(x) + F_7(x))
Eq(F_40(x), F_102(x) + F_41(x))
Eq(F_41(x), F_15(x) + F_31(x) + F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_2(x) + F_44(x))
Eq(F_44(x), F_15(x) + F_45(x) + F_90(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_22(x) + F_4(x))
Eq(F_48(x), F_44(x) + F_49(x))
Eq(F_49(x), F_15(x) + F_50(x) + F_94(x) + F_96(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_59(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_4(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_22(x))
Eq(F_59(x), F_60(x) + F_63(x))
Eq(F_60(x), 2*F_15(x) + F_50(x) + F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_44(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), 2*F_15(x) + F_67(x) + F_94(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_75(x))
Eq(F_69(x), F_44(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))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_2(x) + F_72(x))
Eq(F_75(x), F_66(x) + F_76(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_78(x) + F_81(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_74(x) + F_84(x))
Eq(F_84(x), F_81(x) + F_85(x))
Eq(F_85(x), 2*F_15(x) + F_79(x) + F_86(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_88(x) + F_92(x))
Eq(F_88(x), F_89(x) + F_91(x))
Eq(F_89(x), F_90(x))
Eq(F_90(x), F_2(x)*F_4(x))
Eq(F_91(x), F_71(x))
Eq(F_92(x), F_93(x) + F_95(x))
Eq(F_93(x), 2*F_15(x) + F_86(x) + F_94(x))
Eq(F_94(x), F_4(x)*F_81(x))
Eq(F_95(x), F_77(x))
Eq(F_96(x), F_4(x)*F_97(x))
Eq(F_97(x), F_100(x) + F_98(x))
Eq(F_98(x), F_44(x) + F_99(x))
Eq(F_99(x), F_71(x))
Eq(F_100(x), F_101(x) + F_66(x))
Eq(F_101(x), F_77(x))
Eq(F_102(x), F_103(x) + F_111(x) + F_15(x) + F_96(x))
Eq(F_103(x), F_104(x)*F_4(x))
Eq(F_104(x), F_105(x) + F_108(x))
Eq(F_105(x), F_106(x) + F_107(x))
Eq(F_106(x), F_54(x))
Eq(F_107(x), F_57(x))
Eq(F_108(x), F_109(x) + F_110(x))
Eq(F_109(x), F_103(x) + 2*F_15(x) + F_61(x))
Eq(F_110(x), F_64(x))
Eq(F_111(x), F_112(x)*F_4(x))
Eq(F_112(x), F_66(x) + F_81(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_2(x))
Eq(F_116(x), F_15(x) + F_45(x) + F_73(x))
Eq(F_117(x), F_118(x) + F_81(x))
Eq(F_118(x), 2*F_15(x) + F_67(x) + F_79(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": [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": []}, "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": [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": [[{"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": [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": []}, "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": [2, 0, 1], "pos": [[0, 1], [0, 0], [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": [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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"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": [2, 0, 1], "pos": [[1, 2], [1, 0], [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": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"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", 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