0132_0213_0321_1302_2103
Counting sequence:
1, 1, 2, 6, 19, 55, 152, 417, 1151, 3192, 8862, 24593, 68217, 189196, 524737, 1455425, 4036874, 11196966, 31056599, 86140351, 238923748, 662692625, 1838082663, 5098212736, 14140698342, 39221459781, 108786911585, 301737676728, 836917092305, 2321321710289, 6438552315058, 17858341531446, 49532930172251, 137387403367319, 381065657506192, 1056945773576241, 2931606000898207, 8131272161117832, 22553367313977822, 62555325553044889, 173507073270719865, 481249265491320260, 1334820829895373425, 3702336347679468913, 10269014480709639418, 28482733199299701942, 79001358117320254495, 219122741511848249855, 607771524337143262556, 1685750293404035078209, 4675694628521969450727, 12968777371554255610640, 35970951885303910917510, 99771115076191578901005, 276730942102578196575361, 767556965345121269820912, 2128940445089182771122913, 5904952496520701804394721, 16378318175408333252546018, 45427851691096244536146726, 126001319986982448260757667, 349484557324414872836895799, 969350605381442011872070664, 2688646970118141873118518657, 7457386924600829858210145791, 20684240200104719717597398680, 57371006356697648261969999070, 159127545345541779816973988513, 441365374179833352749639653689, 1224196559444772732615278540284, 3395502465369642672037104464737, 9417962257270540673399477778401, 26122205471500529722397032621418, 72454061723225956828945301998086, 200962781106686015307771564122471, 557402006590164740620303136962687, 1546042482293280689623525854516052, 4288192953731187060769011656181041, 11893980287756108289786190861180839, 32989832456685429828281846417906848, 91502509604839537286071510108147686, 253796659166937115452586392844107861, 703945110166592378902320932623164257, 1952502920070002480299108000033459368, 5415575160369667097051218956274620209, 15020952858068679118869462332457880625, 41662984647584539328704835375513502610, 115558866747421925742812927952038734806, 320520764340454475432995838436458920427, 889014951989230197464645861039952709655, 2465823349968405410103650361337258109440, 6839350428971263914795127446355674617745, 18970018387922541847511501121216722497183, 52616341475026259553250091347313585596584, 145939731507015804419460492556203228950430, 404786890066252595788630763906780047604521, 1122740357800586591100034228205224863838137, 3114097669585771776585172051154131146634676, 8637441620711701794159109009189945837120145, 23957308237261094969197201946480342854264465, 66449377394211108123151304154113130646784602
Generating function in Maple syntax:
-(x-1)^4/(x^6-x^5-3*x^4+9*x^3-9*x^2+5*x-1)
Generating function in latex syntax:
-\frac{\left(x -1\right)^{4}}{x^{6}-x^{5}-3 x^{4}+9 x^{3}-9 x^{2}+5 x -1}
Generating function in sympy syntax:
-(x - 1)**4/(x**6 - x**5 - 3*x**4 + 9*x**3 - 9*x**2 + 5*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^6-x^5-3*x^4+9*x^3-9*x^2+5*x-1)*F(x)+(x-1)^4 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{6}-x^{5}-3 x^{4}+9 x^{3}-9 x^{2}+5 x -1\right) F \! \left(x \right)+\left(x -1\right)^{4} = 0
Explicit closed form in Maple syntax:
8142/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n+4)+8142/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n+4)+8142/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n+4)+8142/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n+4)+8142/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n+4)+8142/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 6)^(-n+4)+9213/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n+3)+9213/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n+3)+9213/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n+3)+9213/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n+3)+9213/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n+3)+9213/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 6)^(-n+3)-22249/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n+2)-22249/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n+2)-22249/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n+2)-22249/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n+2)-22249/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n+2)-22249/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 6)^(-n+2)+9530/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n+1)+9530/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n+1)+9530/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n+1)+9530/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n+1)+9530/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n+1)+9530/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 6)^(-n+1)+8009/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n-1)+8009/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n-1)+8009/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n-1)+8009/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n-1)+8009/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n-1)+8009/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 6)^(-n-1)+27458/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n)+27458/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n)+27458/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n)+27458/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n)+27458/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n)+27458/154361*RootOf(_Z^6-_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 6)^(-n)
Explicit closed form in latex syntax:
\frac{8142 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{154361}+\frac{8142 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{154361}+\frac{8142 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{154361}+\frac{8142 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{154361}+\frac{8142 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{154361}+\frac{8142 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{154361}+\frac{9213 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{154361}+\frac{9213 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{154361}+\frac{9213 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{154361}+\frac{9213 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{154361}+\frac{9213 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{154361}+\frac{9213 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{154361}-\frac{22249 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{154361}-\frac{22249 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{154361}-\frac{22249 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{154361}-\frac{22249 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{154361}-\frac{22249 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{154361}-\frac{22249 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{154361}+\frac{9530 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{154361}+\frac{9530 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{154361}+\frac{9530 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{154361}+\frac{9530 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{154361}+\frac{9530 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{154361}+\frac{9530 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{154361}+\frac{8009 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{154361}+\frac{8009 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{154361}+\frac{8009 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{154361}+\frac{8009 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{154361}+\frac{8009 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{154361}+\frac{8009 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{154361}+\frac{27458 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{154361}+\frac{27458 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{154361}+\frac{27458 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{154361}+\frac{27458 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{154361}+\frac{27458 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{154361}+\frac{27458 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{154361}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(n+6) = a(n)-a(n+1)-3*a(n+2)+9*a(n+3)-9*a(n+4)+5*a(n+5), 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) = 19
a \! \left(5\right) = 55
a \! \left(n +6\right) = a \! \left(n \right)-a \! \left(n +1\right)-3 a \! \left(n +2\right)+9 a \! \left(n +3\right)-9 a \! \left(n +4\right)+5 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18271/
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[16,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[13,x]*F[4,x]
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[18,x]+F[19,x]+F[54,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[33,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[18,x]+F[23,x]+F[31,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[11,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[25,x]*F[4,x]
F[28,x] = F[14,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[28,x]*F[4,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[13,x]
F[33,x] = F[34,x]+F[43,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[2,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[37,x]*F[4,x]
F[40,x] = F[38,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = 2*F[18,x]+F[44,x]+F[52,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[38,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[46,x]
F[49,x] = F[41,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[49,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[40,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[65,x]
F[56,x] = F[2,x]+F[57,x]
F[57,x] = F[18,x]+F[58,x]+F[64,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[62,x]
F[60,x] = F[11,x]+F[61,x]
F[61,x] = F[15,x]+F[18,x]+F[23,x]
F[62,x] = F[38,x]+F[63,x]
F[63,x] = 2*F[18,x]+F[42,x]+F[44,x]
F[64,x] = F[4,x]*F[56,x]
F[65,x] = F[38,x]+F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[65,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_{16}\! \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_{13}\! \left(x \right) F_{4}\! \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_{19}\! \left(x \right)+F_{54}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{33}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{31}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \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_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \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_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{43}\! \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_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{52}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{38}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{41}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{40}\! \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_{65}\! \left(x \right)
F_{56}\! \left(x \right) = F_{2}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{18}\! \left(x \right)+F_{58}\! \left(x \right)+F_{64}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{11}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{23}\! \left(x \right)
F_{62}\! \left(x \right) = F_{38}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{65}\! \left(x \right) = F_{38}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \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_16(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_13(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_54(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_33(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_18(x) + F_23(x) + F_31(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_11(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_25(x)*F_4(x))
Eq(F_28(x), F_14(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_28(x)*F_4(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_13(x))
Eq(F_33(x), F_34(x) + F_43(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_2(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_37(x)*F_4(x))
Eq(F_40(x), F_38(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_40(x))
Eq(F_43(x), 2*F_18(x) + F_44(x) + F_52(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_38(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_46(x))
Eq(F_49(x), F_41(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_49(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_40(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_65(x))
Eq(F_56(x), F_2(x) + F_57(x))
Eq(F_57(x), F_18(x) + F_58(x) + F_64(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_62(x))
Eq(F_60(x), F_11(x) + F_61(x))
Eq(F_61(x), F_15(x) + F_18(x) + F_23(x))
Eq(F_62(x), F_38(x) + F_63(x))
Eq(F_63(x), 2*F_18(x) + F_42(x) + F_44(x))
Eq(F_64(x), F_4(x)*F_56(x))
Eq(F_65(x), F_38(x) + F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_65(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, 1, 3], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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, 1, 3], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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, 1, 3], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"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, 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": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "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, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [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, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 2], [1, 0], [1, 2]]}, {"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, 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": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 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