0132_0321_1203_2031
Counting sequence:
1, 1, 2, 6, 20, 64, 197, 597, 1804, 5454, 16495, 49883, 150827, 456009, 1378707, 4168563, 12604150, 38110748, 115234852, 348434144, 1053555461, 3185619065, 9632301812, 29125017656, 88064777943, 266279823051, 805145303287, 2434502734303, 7361160235983, 22257802202029, 67300499513734, 203495259970302, 615304807182908, 1860485624835856, 5625515549492473, 17009766038364041, 51432111086619912, 155514311291766854, 470225711230434491, 1421812678601568767, 4299108374421395491, 12999133495677617941, 39305236556459241611, 118846507828698789455, 359353960452503486450, 1086571841716084875580, 3285446932945659405164, 9934144374802937940728, 30037686340272471722865, 90824389764769327509701, 274624006745915197846728, 830375467168121403715536, 2510790752217907069614059, 7591830985713110343392927, 22955277202897509365934223, 69409441866331173733340627, 209872029756517170384019383, 634586126754121067490193305, 1918786189546024495051948650, 5801797874190042569665770582, 17542787599967088752712665972, 53043798396806830372059646032, 160387540026213704245956642221, 484960801698712157678493456669, 1466366895744012961106373664276, 4433826126569691051759562081838, 13406477040439048724871505672887, 40536913605783069411762001816147, 122570706661121915234788207239979, 370614750730887055284256701794257, 1120620882435404451581868288838083, 3388400379838542825299576848819979, 10245442784484065949390256341881870, 30978953512907581760784612347115964, 93670481690481354002779692568779012, 283229681611775307589416591358062912, 856396285128310009217690500600264477, 2589469412273201018345330487986693073, 7829730176951765214118657179340554284, 23674608533047693498550196925906297944, 71584470540626080059281170576037163759, 216448623233986545200884811907841048003, 654471649312536892366390371633098477879, 1978913671771582942478550888663955050519, 5983604216375140859648455611714882465887, 18092512032710339411206252174257663907909, 54705989871113154126509764157984352443054, 165413477264384516043574348707775619180046, 500157634020676585613961479066962192712188, 1512317273092130152909981156671968620585536, 4572765422187412611768432201239270720902177, 13826585187114357912915403791494093239271825, 41807186742826754553847321036107284996645744, 126411607761147262295301963315637922660695110, 382228411470426552250967323214932909668536163, 1155736890960651563041269660422451211410673815, 3494579997308074990035031441186306158069631939, 10566496105731283185707998278539968191295959869, 31949716428995934735764723386832743955221888507, 96605759343400295183430244694347721041060016039, 292105025691092368226298767129121639811387318346
Generating function in Maple syntax:
(2*x-1)*(x-1)^5/(5*x^6-26*x^5+47*x^4-46*x^3+26*x^2-8*x+1)
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(x -1\right)^{5}}{5 x^{6}-26 x^{5}+47 x^{4}-46 x^{3}+26 x^{2}-8 x +1}
Generating function in sympy syntax:
(x - 1)**5*(2*x - 1)/(5*x**6 - 26*x**5 + 47*x**4 - 46*x**3 + 26*x**2 - 8*x + 1)
Implicit equation for the generating function in Maple syntax:
(5*x^6-26*x^5+47*x^4-46*x^3+26*x^2-8*x+1)*F(x)-(2*x-1)*(x-1)^5 = 0
Implicit equation for the generating function in latex syntax:
\left(5 x^{6}-26 x^{5}+47 x^{4}-46 x^{3}+26 x^{2}-8 x +1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right)^{5} = 0
Explicit closed form in Maple syntax:
-115/181*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n+4)-115/181*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n+4)-115/181*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n+4)-115/181*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n+4)-115/181*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n+4)-115/181*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n+4)+2447/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n+3)+2447/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n+3)+2447/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n+3)+2447/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n+3)+2447/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n+3)+2447/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n+3)-2205/362*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n+2)-2205/362*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n+2)-2205/362*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n+2)-2205/362*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n+2)-2205/362*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n+2)-2205/362*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n+2)+3811/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n+1)+3811/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n+1)+3811/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n+1)+3811/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n+1)+3811/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n+1)+3811/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n+1)+333/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n-1)+333/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n-1)+333/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n-1)+333/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n-1)+333/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n-1)+333/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n-1)-1681/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n)-1681/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n)-1681/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n)-1681/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n)-1681/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n)-1681/724*RootOf(5*_Z^6-26*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n)+piecewise(n = 0,2/5,0)
Explicit closed form in latex syntax:
-\frac{115 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{181}-\frac{115 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{181}-\frac{115 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{181}-\frac{115 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{181}-\frac{115 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{181}-\frac{115 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{181}+\frac{2447 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{724}+\frac{2447 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{724}+\frac{2447 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{724}+\frac{2447 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{724}+\frac{2447 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{724}+\frac{2447 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{724}-\frac{2205 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{362}-\frac{2205 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{362}-\frac{2205 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{362}-\frac{2205 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{362}-\frac{2205 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{362}-\frac{2205 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{362}+\frac{3811 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{724}+\frac{3811 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{724}+\frac{3811 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{724}+\frac{3811 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{724}+\frac{3811 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{724}+\frac{3811 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{724}+\frac{333 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{724}+\frac{333 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{724}+\frac{333 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{724}+\frac{333 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{724}+\frac{333 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{724}+\frac{333 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{724}-\frac{1681 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{724}-\frac{1681 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{724}-\frac{1681 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{724}-\frac{1681 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{724}-\frac{1681 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{724}-\frac{1681 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-26 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{724}+\left(\left\{\begin{array}{cc}\frac{2}{5} & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(6) = 197
a(n+6) = -5*a(n)+26*a(n+1)-47*a(n+2)+46*a(n+3)-26*a(n+4)+8*a(n+5), 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) = 20
a \! \left(5\right) = 64
a \! \left(6\right) = 197
a \! \left(n +6\right) = -5 a \! \left(n \right)+26 a \! \left(n +1\right)-47 a \! \left(n +2\right)+46 a \! \left(n +3\right)-26 a \! \left(n +4\right)+8 a \! \left(n +5\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15431/
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[19,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[18,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]
F[18,x] = F[12,x]*F[13,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,x]+F[21,x]+F[44,x]
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[30,x]
F[23,x] = F[24,x]+F[6,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[10,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[27,x]
F[30,x] = F[20,x]+F[31,x]
F[31,x] = 2*F[15,x]+F[32,x]+F[36,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[24,x]
F[35,x] = F[31,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[2,x]+F[39,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[38,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[59,x]
F[46,x] = F[2,x]+F[47,x]
F[47,x] = F[15,x]+F[48,x]+F[58,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[50,x]+F[52,x]
F[50,x] = F[10,x]+F[51,x]
F[51,x] = F[29,x]
F[52,x] = F[47,x]+F[53,x]
F[53,x] = 2*F[15,x]+F[43,x]+F[54,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[51,x]
F[57,x] = F[53,x]
F[58,x] = F[12,x]*F[46,x]
F[59,x] = F[39,x]+F[60,x]
F[60,x] = 2*F[15,x]+F[61,x]+F[63,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[47,x]
F[63,x] = F[12,x]*F[59,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_{19}\! \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_{18}\! \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_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{44}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{6}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{20}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{24}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \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_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{2}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{59}\! \left(x \right)
F_{46}\! \left(x \right) = F_{2}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{15}\! \left(x \right)+F_{48}\! \left(x \right)+F_{58}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{10}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{29}\! \left(x \right)
F_{52}\! \left(x \right) = F_{47}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{51}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{59}\! \left(x \right) = F_{39}\! \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_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{47}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \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_19(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_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x))
Eq(F_18(x), F_12(x)*F_13(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_21(x) + F_44(x))
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_30(x))
Eq(F_23(x), F_24(x) + F_6(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_10(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_27(x))
Eq(F_30(x), F_20(x) + F_31(x))
Eq(F_31(x), 2*F_15(x) + F_32(x) + F_36(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_24(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(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_12(x)*F_41(x))
Eq(F_41(x), F_2(x) + F_39(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_38(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_59(x))
Eq(F_46(x), F_2(x) + F_47(x))
Eq(F_47(x), F_15(x) + F_48(x) + F_58(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_52(x))
Eq(F_50(x), F_10(x) + F_51(x))
Eq(F_51(x), F_29(x))
Eq(F_52(x), F_47(x) + F_53(x))
Eq(F_53(x), 2*F_15(x) + F_43(x) + F_54(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_51(x))
Eq(F_57(x), F_53(x))
Eq(F_58(x), F_12(x)*F_46(x))
Eq(F_59(x), F_39(x) + F_60(x))
Eq(F_60(x), 2*F_15(x) + F_61(x) + F_63(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_47(x))
Eq(F_63(x), F_12(x)*F_59(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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 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, 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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 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, 1, 2], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 1], [0, 1], [0, 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}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 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, 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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 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, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 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, 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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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