0123_0213_0321_1032_2103
Counting sequence:
1, 1, 2, 6, 19, 53, 146, 406, 1135, 3171, 8852, 24708, 68973, 192547, 537516, 1500524, 4188845, 11693541, 32643590, 91127562, 254390895, 710155369, 1982463454, 5534227474, 15449300539, 43128130919, 120396109320, 336096715336, 938244621849, 2619195398983, 7311722740824, 20411340619896, 56980118184121, 159065194625769, 444044992317738, 1239592078369806, 3460434296841867, 9660117817554461, 26967099573079274, 75281116971767406, 210154101191305959, 586664332625254795, 1637726969036538972, 4571864140959959884, 12762775553297500293, 35628451502858182123, 99460070514485216772, 277651854332001828644, 775090464095054197925, 2163735693307210164877, 6040265449475947699822, 16861951675977048061970, 47071675359508115621351, 131404873156388915423441, 366828683223377360219318, 1024035711942350116855210, 2858689048300809815464435, 7980291096854884493797711, 22277710137237697063896464, 62190258843362092238947216, 173609777269682157569709361, 484647520756310424735336079, 1352937738122750861513025200, 3776849039442593018239237488, 10543418418153566101713091313, 29432913727646884626275844689, 82164661985483076982322712786, 229370144650249761996826551830, 640307669812745005431658279939, 1787477235305396305925281491461, 4989905661553933707577107285570, 13929776569688116056524213548230, 38886241232265176049747048021215, 108554487547518264261808511747891, 303039748591763275426426041733028, 845963085463052779104806827963668, 2361583077110631056129523399601821, 6592574458544614703158149555467699, 18403772627228557551508078363180636, 51375809108343195070862434160137564, 143420254911853939101573855690856989, 400370716801826347377864604561783221, 1117671356608078557115997990734777686, 3120081486879722654831769906175294234, 8709991919550653326533221525964959327, 24314736508534716537537812898801969465, 67876803668719274016477684172938243214, 189484285575731722649664794601165595842, 528962657926271636237262101553853617963, 1476647483617298722077656017709260439479, 4122196072255110434142945626017089316504, 11507486144553231295357559037899321551832, 32124196677195168097040969315271083587913, 89677623695733959519082545971312321510871, 250343262199695750108742590348718012159752, 698856039511303009407085889682731916734184, 1950920347006716765277060433304435471833769, 5446172008510273486225379072571417337981497, 15203485673717620094003424190116783744354042, 42441916316587981692744173715130799866390046, 118480478706156582516696625435148706414226555
Generating function in Maple syntax:
(x^3+x^2+x-1)/(x^6+2*x^5+3*x^4+2*x^3+x^2+2*x-1)
Generating function in latex syntax:
\frac{x^{3}+x^{2}+x -1}{x^{6}+2 x^{5}+3 x^{4}+2 x^{3}+x^{2}+2 x -1}
Generating function in sympy syntax:
(x**3 + x**2 + x - 1)/(x**6 + 2*x**5 + 3*x**4 + 2*x**3 + x**2 + 2*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^6+2*x^5+3*x^4+2*x^3+x^2+2*x-1)*F(x)-x^3-x^2-x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{6}+2 x^{5}+3 x^{4}+2 x^{3}+x^{2}+2 x -1\right) F \! \left(x \right)-x^{3}-x^{2}-x +1 = 0
Explicit closed form in Maple syntax:
-823/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 1)^(-n+4)-823/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 2)^(-n+4)-823/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 3)^(-n+4)-823/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 4)^(-n+4)-823/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 5)^(-n+4)-823/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 6)^(-n+4)-3163/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 1)^(-n+3)-3163/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 2)^(-n+3)-3163/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 3)^(-n+3)-3163/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 4)^(-n+3)-3163/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 5)^(-n+3)-3163/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 6)^(-n+3)-1391/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 1)^(-n+2)-1391/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 2)^(-n+2)-1391/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 3)^(-n+2)-1391/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 4)^(-n+2)-1391/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 5)^(-n+2)-1391/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 6)^(-n+2)+1559/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 2)^(-n+1)+1559/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 3)^(-n+1)+1559/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 4)^(-n+1)+1559/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 5)^(-n+1)+1559/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 6)^(-n+1)+1559/39898*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 1)^(-n+1)+717/19949*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 1)^(-n-1)+717/19949*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 2)^(-n-1)+717/19949*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 3)^(-n-1)+717/19949*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 4)^(-n-1)+717/19949*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 5)^(-n-1)+717/19949*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 6)^(-n-1)+4031/19949*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 1)^(-n)+4031/19949*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 2)^(-n)+4031/19949*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 3)^(-n)+4031/19949*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 4)^(-n)+4031/19949*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 5)^(-n)+4031/19949*RootOf(_Z^6+2*_Z^5+3*_Z^4+2*_Z^3+_Z^2+2*_Z-1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{823 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{39898}-\frac{823 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{39898}-\frac{823 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{39898}-\frac{823 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{39898}-\frac{823 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{39898}-\frac{823 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{39898}-\frac{3163 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{39898}-\frac{3163 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{39898}-\frac{3163 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{39898}-\frac{3163 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{39898}-\frac{3163 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{39898}-\frac{3163 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{39898}-\frac{1391 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{39898}-\frac{1391 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{39898}-\frac{1391 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{39898}-\frac{1391 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{39898}-\frac{1391 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{39898}-\frac{1391 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{39898}+\frac{1559 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{39898}+\frac{1559 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{39898}+\frac{1559 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{39898}+\frac{1559 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{39898}+\frac{1559 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{39898}+\frac{1559 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{39898}+\frac{717 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{19949}+\frac{717 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{19949}+\frac{717 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{19949}+\frac{717 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{19949}+\frac{717 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{19949}+\frac{717 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{19949}+\frac{4031 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{19949}+\frac{4031 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{19949}+\frac{4031 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{19949}+\frac{4031 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{19949}+\frac{4031 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{19949}+\frac{4031 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{19949}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(n+6) = a(n)+2*a(n+1)+3*a(n+2)+2*a(n+3)+a(n+4)+2*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) = 53
a \! \left(n +6\right) = a \! \left(n \right)+2 a \! \left(n +1\right)+3 a \! \left(n +2\right)+2 a \! \left(n +3\right)+a \! \left(n +4\right)+2 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16822/
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[20,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[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[15,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[64,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[37,x]
F[25,x] = F[15,x]+F[26,x]
F[26,x] = F[22,x]+F[27,x]+F[35,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[30,x]+F[4,x]
F[30,x] = F[31,x]
F[31,x] = x^2
F[32,x] = F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[15,x]*F[4,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[15,x]+F[33,x]
F[37,x] = F[38,x]+F[43,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[2,x]*F[4,x]
F[43,x] = 2*F[22,x]+F[44,x]+F[62,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[60,x]
F[46,x] = F[41,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[22,x]+F[42,x]+F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[54,x]
F[52,x] = F[4,x]+F[53,x]
F[53,x] = F[22,x]+F[27,x]+F[34,x]
F[54,x] = F[41,x]+F[55,x]
F[55,x] = 2*F[22,x]+F[44,x]+F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[22,x]+F[23,x]+F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[2,x]+F[49,x]
F[60,x] = F[61,x]
F[61,x] = F[56,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[57,x]+F[61,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[69,x]
F[66,x] = F[2,x]+F[67,x]
F[67,x] = F[22,x]+F[50,x]+F[68,x]
F[68,x] = F[4,x]*F[59,x]
F[69,x] = F[57,x]+F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[57,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_{20}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{64}\! \left(x \right)
F_{22}\! \left(x \right) = 0
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_{37}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{35}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = x^{2}
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{33}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \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_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{44}\! \left(x \right)+F_{62}\! \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_{60}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \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_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{22}\! \left(x \right)+F_{42}\! \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_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{54}\! \left(x \right) = F_{41}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{44}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{2}\! \left(x \right)+F_{49}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{56}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{69}\! \left(x \right)
F_{66}\! \left(x \right) = F_{2}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{22}\! \left(x \right)+F_{50}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{69}\! \left(x \right) = F_{57}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{57}\! \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_20(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_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_15(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_64(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_37(x))
Eq(F_25(x), F_15(x) + F_26(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_35(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_30(x) + F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), x**2)
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_15(x)*F_4(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_15(x) + F_33(x))
Eq(F_37(x), F_38(x) + F_43(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_2(x)*F_4(x))
Eq(F_43(x), 2*F_22(x) + F_44(x) + F_62(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_60(x))
Eq(F_46(x), F_41(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_22(x) + F_42(x) + F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_54(x))
Eq(F_52(x), F_4(x) + F_53(x))
Eq(F_53(x), F_22(x) + F_27(x) + F_34(x))
Eq(F_54(x), F_41(x) + F_55(x))
Eq(F_55(x), 2*F_22(x) + F_44(x) + F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_22(x) + F_23(x) + F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_2(x) + F_49(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_56(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_57(x) + F_61(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_69(x))
Eq(F_66(x), F_2(x) + F_67(x))
Eq(F_67(x), F_22(x) + F_50(x) + F_68(x))
Eq(F_68(x), F_4(x)*F_59(x))
Eq(F_69(x), F_57(x) + F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_57(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, 2, 3], "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, 0, 3, 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, 2, 3], "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, 0, 3, 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, 2, 3], "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, 0, 3, 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, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [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, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "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": [1, 0, 3, 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": []}], "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, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [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, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "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": [1, 0, 3, 2], "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, 1, 2], "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, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, 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