0231_1302_2013
Counting sequence:
1, 1, 2, 6, 21, 76, 274, 977, 3449, 12086, 42141, 146469, 508098, 1760610, 6096937, 21106816, 73058238, 252865657, 875192717, 3029137402, 10484276121, 36287944313, 125600101410, 434730756798, 1504709415309, 5208181828372, 18026872786362, 62395762081153, 215968314664513, 747523953191758, 2587380192399925, 8955614897066173, 30997779065098274, 107291606092059034, 371364949247890753, 1285393418536056392, 4449090393420551366, 15399491651849174377, 53301758616515551381, 184491639634495247154, 638574897428171677361, 2210278471441478370993, 7650364802059660761026, 26479958224581435757174, 91654215926143552046917, 317239748843685032691548, 1098051597819240766955170, 3800650189225708840256945, 13155066564846247653020809, 45533202928642828435952294, 157602590510231950184377421, 545504707292036889601467925, 1888137655069075822834385666, 6535349295509065436033251538, 22620591406408527963305690777, 78295915403900540695111882384, 271003098849173711139894281486, 938014188952379017280597791449, 3246717924675064311579381041437, 11237758881003549382221770306986, 38896888364644587469338084169545, 134632531314510298396625983547049, 465999190429538512545538053474018, 1612947802145198304156452028411694, 5582843631223833168396319752944957, 19323714610753789157117856897671972, 66884543258468945050010990854222282, 231505288553808147064300670981854305, 801301706154593858141978028610923601, 2773519466001419830832926712359498046, 9599892486444691525840143103064062181, 33227794822064552292339835462265094701, 115010282698084337992154252969433808226, 398081341152070385337399642388910813962, 1377865965162700789512071406765952004145, 4769162534620010223672602652242932827992, 16507346764267743516567947498133172696726, 57136341069888063415328992511961854753993, 197764154195944859395520387178856721356581, 684514618060649943424687252311711600366434, 2369288126272204224755074343513106998436705, 8200739731751174904383900496942200426654817, 28384953016978834120903228368811435940326786, 98247912277548471776899822630762810017114342, 340062295016764637405925261363070555302313845, 1177046532707805544046967344247232588195036268, 4074072781550706154028136820675066642224170290, 14101455268032869620839723911997818480561508753, 48808907287278186007755495234790862199839797657, 168940679192073624385275506017961700289487299350, 584748863929559930123558399645088976903300730365, 2023972174742823917885848295925227620451572501253, 7005508889071847247173909481674615199569502253890, 24247939476292779297793566381615146652184581935170, 83928602212344963157179689867142351602848427382281, 290499334024029863225189658134039619277685391298208, 1005495872013844465431104253596477044635143248889694, 3480290073757107461131799485063114613422522486504249, 12046214544106531241015010144043849121900874970494637, 41695169588548259010922656992072545948536308331212890, 144318130866126237991990576038492660890808296375103737
Generating function in Maple syntax:
-(2*x-1)*(x^2-3*x+1)^2/(x-1)/(x^5-8*x^4+24*x^3-22*x^2+8*x-1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x^{2}-3 x +1\right)^{2}}{\left(x -1\right) \left(x^{5}-8 x^{4}+24 x^{3}-22 x^{2}+8 x -1\right)}
Generating function in sympy syntax:
(1 - 2*x)*(x**2 - 3*x + 1)**2/((x - 1)*(x**5 - 8*x**4 + 24*x**3 - 22*x**2 + 8*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^5-8*x^4+24*x^3-22*x^2+8*x-1)*F(x)+(2*x-1)*(x^2-3*x+1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{5}-8 x^{4}+24 x^{3}-22 x^{2}+8 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x^{2}-3 x +1\right)^{2} = 0
Explicit closed form in Maple syntax:
1/2+5341/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 1)^(-n+4)+5341/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 2)^(-n+4)+5341/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 3)^(-n+4)+5341/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 4)^(-n+4)+5341/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 5)^(-n+4)-44137/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 1)^(-n+3)-44137/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 2)^(-n+3)-44137/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 3)^(-n+3)-44137/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 4)^(-n+3)-44137/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 5)^(-n+3)+138343/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 1)^(-n+2)+138343/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 2)^(-n+2)+138343/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 3)^(-n+2)+138343/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 4)^(-n+2)+138343/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 5)^(-n+2)-143489/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 1)^(-n+1)-143489/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 2)^(-n+1)-143489/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 3)^(-n+1)-143489/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 4)^(-n+1)-143489/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 5)^(-n+1)-3276/4757*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 1)^(-n-1)-3276/4757*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 2)^(-n-1)-3276/4757*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 3)^(-n-1)-3276/4757*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 4)^(-n-1)-3276/4757*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 5)^(-n-1)+55251/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 1)^(-n)+55251/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 2)^(-n)+55251/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 3)^(-n)+55251/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 4)^(-n)+55251/9514*RootOf(_Z^5-8*_Z^4+24*_Z^3-22*_Z^2+8*_Z-1,index = 5)^(-n)
Explicit closed form in latex syntax:
\frac{1}{2}+\frac{5341 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{9514}+\frac{5341 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{9514}+\frac{5341 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{9514}+\frac{5341 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{9514}+\frac{5341 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{9514}-\frac{44137 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{9514}-\frac{44137 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{9514}-\frac{44137 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{9514}-\frac{44137 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{9514}-\frac{44137 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{9514}+\frac{138343 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{9514}+\frac{138343 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{9514}+\frac{138343 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{9514}+\frac{138343 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{9514}+\frac{138343 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{9514}-\frac{143489 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{9514}-\frac{143489 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{9514}-\frac{143489 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{9514}-\frac{143489 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{9514}-\frac{143489 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{9514}-\frac{3276 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{4757}-\frac{3276 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{4757}-\frac{3276 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{4757}-\frac{3276 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{4757}-\frac{3276 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{4757}+\frac{55251 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{9514}+\frac{55251 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{9514}+\frac{55251 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{9514}+\frac{55251 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{9514}+\frac{55251 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-8 \textit{\_Z}^{4}+24 \textit{\_Z}^{3}-22 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{9514}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 76
a(n+5) = a(n)-8*a(n+1)+24*a(n+2)-22*a(n+3)+8*a(n+4)-1, n >= 6
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) = 76
a \! \left(n +5\right) = a \! \left(n \right)-8 a \! \left(n +1\right)+24 a \! \left(n +2\right)-22 a \! \left(n +3\right)+8 a \! \left(n +4\right)-1, \quad n \geq 6
Specification 1
Strategy pack name: insertion_point_placements
Tree: http://permpal.com/tree/370/
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[8,x]
F[4,x] = F[16,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[5,x]
F[10,x] = F[11,x]+F[6,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[14,x]+F[15,x]
F[14,x] = F[6,x]
F[15,x] = F[11,x]
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[8,x]
F[19,x] = F[20,x]+F[70,x]
F[20,x] = F[21,x]+F[68,x]
F[21,x] = F[22,x]+F[45,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[8,x]
F[24,x] = F[25,x]+F[31,x]
F[25,x] = F[1,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[25,x]*F[28,x]*F[8,x]
F[28,x] = F[1,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[28,x]*F[8,x]
F[31,x] = F[29,x]+F[32,x]
F[32,x] = F[33,x]+F[34,x]+F[36,x]
F[33,x] = 0
F[34,x] = F[35,x]*F[8,x]
F[35,x] = F[26,x]+F[32,x]
F[36,x] = F[37,x]*F[8,x]
F[37,x] = F[31,x]+F[38,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[40,x]
F[40,x] = F[41,x]*F[8,x]
F[41,x] = F[29,x]+F[39,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]*F[8,x]
F[44,x] = F[32,x]+F[42,x]
F[45,x] = F[46,x]
F[46,x] = F[47,x]*F[8,x]
F[47,x] = F[48,x]+F[66,x]
F[48,x] = F[49,x]+F[51,x]
F[49,x] = F[50,x]+F[6,x]
F[50,x] = F[28,x]*F[6,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[22,x]*F[6,x]
F[53,x] = F[45,x]+F[54,x]
F[54,x] = F[55,x]*F[6,x]
F[55,x] = F[56,x]
F[56,x] = F[57,x]*F[8,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[22,x]+F[55,x]
F[59,x] = F[22,x]*F[60,x]
F[60,x] = F[28,x]+F[61,x]
F[61,x] = F[29,x]+F[62,x]
F[62,x] = F[33,x]+F[63,x]+F[65,x]
F[63,x] = F[64,x]*F[8,x]
F[64,x] = F[29,x]+F[62,x]
F[65,x] = F[61,x]*F[8,x]
F[66,x] = F[67,x]
F[67,x] = F[22,x]*F[28,x]*F[49,x]
F[68,x] = F[69,x]
F[69,x] = F[22,x]*F[26,x]*F[5,x]
F[70,x] = F[2,x]*F[71,x]
F[71,x] = F[20,x]+F[72,x]
F[72,x] = F[73,x]
F[73,x] = F[25,x]*F[5,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{16}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{5}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{6}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{6}\! \left(x \right)
F_{15}\! \left(x \right) = F_{11}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{8}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{70}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{68}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{45}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{8}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{31}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right) F_{28}\! \left(x \right) F_{8}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right) F_{8}\! \left(x \right)
F_{31}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = 0
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{8}\! \left(x \right)
F_{35}\! \left(x \right) = F_{26}\! \left(x \right)+F_{32}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{8}\! \left(x \right)
F_{37}\! \left(x \right) = F_{31}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right) F_{8}\! \left(x \right)
F_{41}\! \left(x \right) = F_{29}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right) F_{8}\! \left(x \right)
F_{44}\! \left(x \right) = F_{32}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right) F_{8}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{66}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{6}\! \left(x \right)
F_{50}\! \left(x \right) = F_{28}\! \left(x \right) F_{6}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{22}\! \left(x \right) F_{6}\! \left(x \right)
F_{53}\! \left(x \right) = F_{45}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right) F_{6}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right) F_{8}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{22}\! \left(x \right)+F_{55}\! \left(x \right)
F_{59}\! \left(x \right) = F_{22}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{28}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{29}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{33}\! \left(x \right)+F_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right) F_{8}\! \left(x \right)
F_{64}\! \left(x \right) = F_{29}\! \left(x \right)+F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{61}\! \left(x \right) F_{8}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{22}\! \left(x \right) F_{28}\! \left(x \right) F_{49}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{22}\! \left(x \right) F_{26}\! \left(x \right) F_{5}\! \left(x \right)
F_{70}\! \left(x \right) = F_{2}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{20}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{25}\! \left(x \right) F_{5}\! \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_8(x))
Eq(F_4(x), F_16(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_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_5(x))
Eq(F_10(x), F_11(x) + F_6(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_8(x))
Eq(F_13(x), F_14(x) + F_15(x))
Eq(F_14(x), F_6(x))
Eq(F_15(x), F_11(x))
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_8(x))
Eq(F_19(x), F_20(x) + F_70(x))
Eq(F_20(x), F_21(x) + F_68(x))
Eq(F_21(x), F_22(x) + F_45(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_8(x))
Eq(F_24(x), F_25(x) + F_31(x))
Eq(F_25(x), F_1(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_25(x)*F_28(x)*F_8(x))
Eq(F_28(x), F_1(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_28(x)*F_8(x))
Eq(F_31(x), F_29(x) + F_32(x))
Eq(F_32(x), F_33(x) + F_34(x) + F_36(x))
Eq(F_33(x), 0)
Eq(F_34(x), F_35(x)*F_8(x))
Eq(F_35(x), F_26(x) + F_32(x))
Eq(F_36(x), F_37(x)*F_8(x))
Eq(F_37(x), F_31(x) + F_38(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_41(x)*F_8(x))
Eq(F_41(x), F_29(x) + F_39(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x)*F_8(x))
Eq(F_44(x), F_32(x) + F_42(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_47(x)*F_8(x))
Eq(F_47(x), F_48(x) + F_66(x))
Eq(F_48(x), F_49(x) + F_51(x))
Eq(F_49(x), F_50(x) + F_6(x))
Eq(F_50(x), F_28(x)*F_6(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_22(x)*F_6(x))
Eq(F_53(x), F_45(x) + F_54(x))
Eq(F_54(x), F_55(x)*F_6(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_57(x)*F_8(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_22(x) + F_55(x))
Eq(F_59(x), F_22(x)*F_60(x))
Eq(F_60(x), F_28(x) + F_61(x))
Eq(F_61(x), F_29(x) + F_62(x))
Eq(F_62(x), F_33(x) + F_63(x) + F_65(x))
Eq(F_63(x), F_64(x)*F_8(x))
Eq(F_64(x), F_29(x) + F_62(x))
Eq(F_65(x), F_61(x)*F_8(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_22(x)*F_28(x)*F_49(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_22(x)*F_26(x)*F_5(x))
Eq(F_70(x), F_2(x)*F_71(x))
Eq(F_71(x), F_20(x) + F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_25(x)*F_5(x))
Pack JSON:
{"expansion_strats": [[{"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}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "insertion_point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 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": [[0, 1], [0, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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]]}]]}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 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, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}], [{"patt": [0], "pos": [[1, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], 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