0132_0231_0321_1023_2013
Counting sequence:
1, 1, 2, 6, 19, 55, 159, 464, 1356, 3958, 11550, 33709, 98385, 287148, 838069, 2445988, 7138867, 20835514, 60810572, 177481855, 517998900, 1511832636, 4412437776, 12878149775, 37586193866, 109699141111, 320168134157, 934443361091, 2727268275429, 7959810680755, 23231519482064, 67803559543229, 197891605423471, 577566838096638, 1685687736754210, 4919851623074099, 14359088854541605, 41908465646732303, 122314132230461074, 356986272640054018, 1041900854214633092, 3040893931256958104, 8875159151420078027, 25903057371842135105, 75600715408194822570, 220648400232624813671, 643984865253527008576, 1879535524564767914275, 5485616166942379049056, 16010330391593853241146, 46727782521989618697837, 136379800167571001856176, 398038359406288704487764, 1161715557319925823194880, 3390585365019026534265919, 9895769274195312571736744, 28881812131445480529965661, 84294515048092819205399193, 246022141368922702940776445, 718040717230742536714332289, 2095675083276740713571247725, 6116441518253972348556397308, 17851458532267491289261986029, 52101302820963802780145612073, 152062967333177859925144560242, 443811282678081991208797190294, 1295308503357009238202243823507, 3780490006347997754739107026755, 11033745745555362065577670585515, 32203112552377011049682045006856, 93988069144950593839940125121776, 274313767876576483117830534732750, 800612715328740241923754132648946, 2336669883206367079317416487878981, 6819809426636087545054414480568417, 19904309526091001928971225628735136, 58092757865501816159300018979311149, 169549640091564248476431322625565180, 494847921004802653593673398147544099, 1444264138753302259007105720923856134, 4215232223777643434610835203225730676, 12302585256815295784329877988694678123, 35906350105088086841510252280227994880, 104796345723752751214835981581217079904, 305858825664825099547261177923114475620, 892680184513936373459687099534051284163, 2605378184172762314318652458133006853434, 7604062014952665432964118360012187312403, 22193230709654179079825502017313555047817, 64773207841204507709062118529805428346459, 189047214843523500854148055650726060185637, 551753581939454256131678518358879042765055, 1610349115351949796605873120142823292537860, 4699968170934992464023473569883534828518441, 13717336568334255100872990563346459910386751, 40035446131867603187084774823149644470296950, 116847533702551034928902475647653487407313786, 341031447168037054468786233696244449108398491, 995335068462694597488475292080842073940343897, 2904986935188682155940656378101391932087170971, 8478500719010109411012884010440326463474982834
Generating function in Maple syntax:
-(x^3+2*x-1)/(x^5-x^4-2*x^3+x^2-3*x+1)
Generating function in latex syntax:
-\frac{x^{3}+2 x -1}{x^{5}-x^{4}-2 x^{3}+x^{2}-3 x +1}
Generating function in sympy syntax:
(-x**3 - 2*x + 1)/(x**5 - x**4 - 2*x**3 + x**2 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^5-x^4-2*x^3+x^2-3*x+1)*F(x)+x^3+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}-x^{4}-2 x^{3}+x^{2}-3 x +1\right) F \! \left(x \right)+x^{3}+2 x -1 = 0
Explicit closed form in Maple syntax:
-278/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+3)-278/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+3)-278/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+3)-278/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+3)-278/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+3)+65/2669*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+2)+65/2669*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+2)+65/2669*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+2)+65/2669*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+2)+65/2669*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+2)+826/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+1)+826/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+1)+826/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+1)+826/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+1)+826/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+1)+131/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n-1)+131/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n-1)+131/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n-1)+131/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n-1)+131/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n-1)+1385/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n)+1385/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n)+1385/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n)+1385/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n)+1385/8007*RootOf(_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n)
Explicit closed form in latex syntax:
-\frac{278 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{8007}-\frac{278 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{8007}-\frac{278 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{8007}-\frac{278 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{8007}-\frac{278 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{8007}+\frac{65 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{2669}+\frac{65 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{2669}+\frac{65 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{2669}+\frac{65 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{2669}+\frac{65 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{2669}+\frac{826 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{8007}+\frac{826 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{8007}+\frac{826 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{8007}+\frac{826 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{8007}+\frac{826 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{8007}+\frac{131 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{8007}+\frac{131 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{8007}+\frac{131 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{8007}+\frac{131 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{8007}+\frac{131 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{8007}+\frac{1385 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{8007}+\frac{1385 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{8007}+\frac{1385 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{8007}+\frac{1385 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{8007}+\frac{1385 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{8007}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+5) = -a(n)+a(n+1)+2*a(n+2)-a(n+3)+3*a(n+4), n >= 5
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(n +5\right) = -a \! \left(n \right)+a \! \left(n +1\right)+2 a \! \left(n +2\right)-a \! \left(n +3\right)+3 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18639/
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[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[11,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[65,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[29,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[1,x]+F[4,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[11,x]
F[29,x] = F[30,x]+F[53,x]
F[30,x] = F[19,x]+F[20,x]+F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[2,x]+F[33,x]
F[33,x] = F[19,x]+F[34,x]+F[52,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[37,x]+F[4,x]
F[37,x] = F[38,x]
F[38,x] = F[11,x]*F[4,x]
F[39,x] = F[33,x]+F[40,x]
F[40,x] = 2*F[19,x]+F[41,x]+F[48,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[45,x]
F[43,x] = F[44,x]
F[44,x] = x^2
F[45,x] = F[46,x]
F[46,x] = 2*F[19,x]+F[41,x]+F[47,x]
F[47,x] = F[33,x]*F[4,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[19,x]+F[34,x]+F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[2,x]+F[49,x]
F[52,x] = F[2,x]*F[4,x]
F[53,x] = 2*F[19,x]+F[54,x]+F[63,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[59,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[4,x]
F[59,x] = F[60,x]
F[60,x] = 2*F[19,x]+F[54,x]+F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[33,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[49,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[51,x]+F[67,x]
F[67,x] = F[49,x]+F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[33,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_{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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{65}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{29}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \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_{11}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{53}\! \left(x \right)
F_{30}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \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_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{19}\! \left(x \right)+F_{34}\! \left(x \right)+F_{52}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
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_{38}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{41}\! \left(x \right)+F_{48}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = x^{2}
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{41}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{19}\! \left(x \right)+F_{34}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{2}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{54}\! \left(x \right)+F_{63}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{59}\! \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_{4}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{54}\! \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_{33}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{49}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{51}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{49}\! \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_{33}\! \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_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_11(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_65(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_29(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_1(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_11(x))
Eq(F_29(x), F_30(x) + F_53(x))
Eq(F_30(x), F_19(x) + F_20(x) + F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_2(x) + F_33(x))
Eq(F_33(x), F_19(x) + F_34(x) + F_52(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_37(x) + F_4(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_11(x)*F_4(x))
Eq(F_39(x), F_33(x) + F_40(x))
Eq(F_40(x), 2*F_19(x) + F_41(x) + F_48(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_45(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), x**2)
Eq(F_45(x), F_46(x))
Eq(F_46(x), 2*F_19(x) + F_41(x) + F_47(x))
Eq(F_47(x), F_33(x)*F_4(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_19(x) + F_34(x) + F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_2(x) + F_49(x))
Eq(F_52(x), F_2(x)*F_4(x))
Eq(F_53(x), 2*F_19(x) + F_54(x) + F_63(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_59(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_4(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), 2*F_19(x) + F_54(x) + F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_33(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_49(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_51(x) + F_67(x))
Eq(F_67(x), F_49(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_33(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, 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": [1, 0, 2, 3], "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, 1, 3, 2], "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": [1, 0, 2, 3], "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, 1, 3, 2], "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": [1, 0, 2, 3], "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, 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, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "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, 3, 2], "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": [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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "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, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [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": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "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", "obstructions": [{"patt": [0, 2, 1], "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]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [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, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "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, 3, 2], "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": [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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": 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"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], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "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]]}, {"patt": [1, 0, 2, 3], "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, 1], "pos": [[0, 0], [0, 0], 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