0132_0213_1302_2031_2103
Counting sequence:
1, 1, 2, 6, 19, 56, 159, 449, 1274, 3628, 10342, 29478, 84006, 239389, 682200, 1944167, 5540670, 15790398, 45001233, 128249624, 365500630, 1041646641, 2968608033, 8460292482, 24111150636, 68714836519, 195831753352, 558104745503, 1590553651807, 4532950052444, 12918543318868, 36816810221380, 104924950362412, 299027676487272, 852204847678883, 2428715331996446, 6921643523535609, 19726127817212282, 56217879085383978, 160216437722875995, 456603972519169378, 1301284629000978888, 3708556621485570825, 10569088351840767037, 30121052471474208266, 85842579017910124066, 244644451897185970171, 697217028295553162696, 1987012502329968446803, 5662825955453218838473, 16138598908740729793146, 45993709993223944128992, 131078377429351094881530, 373562842224358565562910, 1064624080855414018779766, 3034093077320882968844569, 8646921450855994608304640, 24643031268933648208072259, 70230658804183415302157458, 200151733861069753389141502, 570416357324647765111616401, 1625640779756476961745097408, 4632945585926055810752682438, 13203522616704465032574382225, 37628978423449321426224486517, 107239564644743747740479347554, 305624141473569598486096853680, 871004243264938536180464154511, 2482292099464715245924843589576, 7074333006654113793431377943163, 20161280576056242454374866320847, 57458029482656593250580228330324, 163750766702321625045479126587352, 466676526101402859112815894684552, 1329990597295845238865659320496756, 3790366324341338733949051977614336, 10802239430798832599905603976690975, 30785514310567940356630568477065174, 87736241863331562710170334840434581, 250041238832205994506106867245321318, 712597437375239495102832188731236386, 2030845432238967843617618014625792459, 5787746283283487501726468467299763922, 16494611804469490390873139418819972336, 47008318136881603735925636789260456017, 133969929104939263952453515409233328361, 381803532134899459759311700961500751122, 1088109384879197827082712594114203844462, 3101024306511286451918946539438636498707, 8837670075459752435056552221373105747736, 25186649520508206436019274847784735695495, 71779927136046122301859875416647256100305, 204567024107783280702944383908682294444410, 582999579715364628727525671409729564211028, 1661501952383144924417642763862141523775614, 4735147046110723992775262379024974507482166, 13494788565328543860621126455748582553248726, 38459062970916590640001582292876666067261493, 109605238899489013731840442504342028379889928, 312366122994175169874600901853508433638482575, 890218348813498723600996305859474646977539654
Generating function in Maple syntax:
-(2*x-1)*(x-1)^3/(x^6-x^5-7*x^4+14*x^3-13*x^2+6*x-1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x -1\right)^{3}}{x^{6}-x^{5}-7 x^{4}+14 x^{3}-13 x^{2}+6 x -1}
Generating function in sympy syntax:
(1 - 2*x)*(x - 1)**3/(x**6 - x**5 - 7*x**4 + 14*x**3 - 13*x**2 + 6*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^6-x^5-7*x^4+14*x^3-13*x^2+6*x-1)*F(x)+(2*x-1)*(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{6}-x^{5}-7 x^{4}+14 x^{3}-13 x^{2}+6 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
122206/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 1)^(-n+4)+122206/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 2)^(-n+4)+122206/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 3)^(-n+4)+122206/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 4)^(-n+4)+122206/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 5)^(-n+4)+122206/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 6)^(-n+4)-46422/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 1)^(-n+3)-46422/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 2)^(-n+3)-46422/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 3)^(-n+3)-46422/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 4)^(-n+3)-46422/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 5)^(-n+3)-46422/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 6)^(-n+3)-896809/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 1)^(-n+2)-896809/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 2)^(-n+2)-896809/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 3)^(-n+2)-896809/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 4)^(-n+2)-896809/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 5)^(-n+2)-896809/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 6)^(-n+2)+1131010/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 1)^(-n+1)+1131010/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 2)^(-n+1)+1131010/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 3)^(-n+1)+1131010/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 4)^(-n+1)+1131010/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 5)^(-n+1)+1131010/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 6)^(-n+1)+222463/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 1)^(-n-1)+222463/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 2)^(-n-1)+222463/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 3)^(-n-1)+222463/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 4)^(-n-1)+222463/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 5)^(-n-1)+222463/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 6)^(-n-1)-707938/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 1)^(-n)-707938/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 2)^(-n)-707938/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 3)^(-n)-707938/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 4)^(-n)-707938/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 5)^(-n)-707938/725803*RootOf(_Z^6-_Z^5-7*_Z^4+14*_Z^3-13*_Z^2+6*_Z-1,index = 6)^(-n)
Explicit closed form in latex syntax:
\frac{122206 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{725803}+\frac{122206 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{725803}+\frac{122206 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{725803}+\frac{122206 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{725803}+\frac{122206 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{725803}+\frac{122206 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{725803}-\frac{46422 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{725803}-\frac{46422 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{725803}-\frac{46422 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{725803}-\frac{46422 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{725803}-\frac{46422 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{725803}-\frac{46422 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{725803}-\frac{896809 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{725803}-\frac{896809 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{725803}-\frac{896809 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{725803}-\frac{896809 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{725803}-\frac{896809 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{725803}-\frac{896809 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{725803}+\frac{1131010 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{725803}+\frac{1131010 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{725803}+\frac{1131010 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{725803}+\frac{1131010 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{725803}+\frac{1131010 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{725803}+\frac{1131010 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{725803}+\frac{222463 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{725803}+\frac{222463 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{725803}+\frac{222463 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{725803}+\frac{222463 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{725803}+\frac{222463 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{725803}+\frac{222463 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{725803}-\frac{707938 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{725803}-\frac{707938 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{725803}-\frac{707938 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{725803}-\frac{707938 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{725803}-\frac{707938 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{725803}-\frac{707938 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}-7 \textit{\_Z}^{4}+14 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{725803}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(n+6) = a(n)-a(n+1)-7*a(n+2)+14*a(n+3)-13*a(n+4)+6*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) = 56
a \! \left(n +6\right) = a \! \left(n \right)-a \! \left(n +1\right)-7 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18513/
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[14,x]+F[7,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[44,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[28,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,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[29,x]+F[38,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[2,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[32,x]*F[4,x]
F[35,x] = F[29,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[35,x]*F[4,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[33,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[41,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[55,x]
F[46,x] = F[2,x]+F[47,x]
F[47,x] = F[18,x]+F[48,x]+F[54,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[52,x]
F[50,x] = F[11,x]+F[51,x]
F[51,x] = F[23,x]
F[52,x] = F[33,x]+F[53,x]
F[53,x] = F[39,x]
F[54,x] = F[4,x]*F[46,x]
F[55,x] = F[29,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[55,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_{14}\! \left(x \right)+F_{7}\! \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_{44}\! \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_{28}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \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_{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_{29}\! \left(x \right)+F_{38}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{29}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{33}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \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_{55}\! \left(x \right)
F_{46}\! \left(x \right) = F_{2}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \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_{11}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{23}\! \left(x \right)
F_{52}\! \left(x \right) = F_{33}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{39}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{55}\! \left(x \right) = F_{29}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \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_14(x) + F_7(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_44(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_28(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(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_29(x) + F_38(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_2(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_32(x)*F_4(x))
Eq(F_35(x), F_29(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_35(x)*F_4(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_33(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_41(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_55(x))
Eq(F_46(x), F_2(x) + F_47(x))
Eq(F_47(x), F_18(x) + F_48(x) + F_54(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_52(x))
Eq(F_50(x), F_11(x) + F_51(x))
Eq(F_51(x), F_23(x))
Eq(F_52(x), F_33(x) + F_53(x))
Eq(F_53(x), F_39(x))
Eq(F_54(x), F_4(x)*F_46(x))
Eq(F_55(x), F_29(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_55(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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 0], [0, 1], [0, 0]]}, {"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": [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, 1]]}, {"patt": [0, 2, 1, 3], "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, 3, 1], "pos": [[0, 0], [0, 0], [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, 0], [1, 2], [1, 0]]}, {"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": [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, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [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]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"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": [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, 1]]}, {"patt": [0, 2, 1, 3], "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, 3, 1], "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, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"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": [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, 1]]}, {"patt": [0, 2, 1, 3], "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, 3, 1], "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], "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, 0, 2], "pos": [[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], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[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, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"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, 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