0231_0312_0321_2103
Counting sequence:
1, 1, 2, 6, 20, 64, 193, 565, 1646, 4819, 14177, 41788, 123170, 362838, 1068452, 3145951, 9263267, 27277196, 80324321, 236535313, 696534586, 2051102682, 6039922640, 17785881508, 52374471445, 154228297601, 454159616642, 1337374221603, 3938196288881, 11596895900556, 34149642201518, 100561225910026, 296124923949960, 872005784762843, 2567815226057623, 7561503773072776, 22266531762644865, 65568761397207297, 193081819709028642, 568572416339099438, 1674288097740273116, 4930314158200915048, 14518407991574841193, 42752685497407640029, 125894803224880430374, 370725284146089308435, 1091683157562684620257, 3214704169022242835212, 9466412321870234011002, 27875959197486175057934, 82086969673253804551212, 241723362500301181873911, 711808271288763790683419, 2096077970429811650878196, 6172368368474350424984737, 18175913212012117398765585, 53523024124409790877034298, 157610463804862119704813122, 464119109619867221502259640, 1366702074939857180162730380, 4024558616374959703077349181, 11851209092040557491132821513, 34898524367815851248680318810, 102766476702280160016627592291, 302618776154897212957059032561, 891128377854136069561254794684, 2624125957770955066113474908230, 7727323260458966099220943315042, 22754824170998145225101300055568, 67006647140356290433810666706483, 197315994500838536550591878173551, 581040886948159040290898123795152, 1711004286092319773559996210953473, 5038433151241361887911650437058945, 14836788443999392369992605174379778, 43690227641059111226582402078470998, 128655604852246673025643211023001572, 378855079352847885705293333246439952, 1115623150008036574974536608754814993, 3285200808076478021636842782156204229, 9674005374761717979280248280275324958, 28487263171505940405647200758341531859, 83887090358643456471439029006956173281, 247024218734986615364649298954114777500, 727417822942086440513857495841144472018, 2142043771430666432671057958931147932934, 6307724905841654008669781917138911702964, 18574500679414776059315875783260912540559, 54696753685320117084511601033698914899667, 161066771879804843336529279045251916592668, 474296978446523716407921688911732006462305, 1396673076252968688057355365425871895879313, 4112814904111536404399611922720267870643450, 12111099385450068965351404112120982314229418, 35663829212838123601253781037799860444157408, 105020087247448481976390913744007983903389940, 309255034271290109176021015047598023431954277, 910670317734480949758036381911510355452498641, 2681671551626576013044569925497818986261299314, 7896779076640590606395577810556585185483260963, 23253824558583434847420187590862783074319099633
Generating function in Maple syntax:
(x^2+x-1)*(x-1)^2/(x^6+2*x^5+4*x^4+x^3-4*x^2+4*x-1)
Generating function in latex syntax:
\frac{\left(x^{2}+x -1\right) \left(x -1\right)^{2}}{x^{6}+2 x^{5}+4 x^{4}+x^{3}-4 x^{2}+4 x -1}
Generating function in sympy syntax:
(x - 1)**2*(x**2 + x - 1)/(x**6 + 2*x**5 + 4*x**4 + x**3 - 4*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^6+2*x^5+4*x^4+x^3-4*x^2+4*x-1)*F(x)-(x^2+x-1)*(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{6}+2 x^{5}+4 x^{4}+x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x^{2}+x -1\right) \left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
-11423/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 1)^(-n+4)-11423/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 2)^(-n+4)-11423/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 3)^(-n+4)-11423/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 4)^(-n+4)-11423/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 5)^(-n+4)-11423/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 6)^(-n+4)+113561/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 1)^(-n+3)+113561/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 2)^(-n+3)+113561/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 3)^(-n+3)+113561/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 4)^(-n+3)+113561/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 5)^(-n+3)+113561/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 6)^(-n+3)-282792/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 1)^(-n+2)-282792/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 2)^(-n+2)-282792/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 3)^(-n+2)-282792/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 4)^(-n+2)-282792/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 5)^(-n+2)-282792/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 6)^(-n+2)+305838/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 1)^(-n+1)+305838/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 2)^(-n+1)+305838/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 3)^(-n+1)+305838/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 4)^(-n+1)+305838/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 5)^(-n+1)+305838/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 6)^(-n+1)+1416856/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 1)^(-n-1)+1416856/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 2)^(-n-1)+1416856/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 3)^(-n-1)+1416856/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 4)^(-n-1)+1416856/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 5)^(-n-1)+1416856/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 6)^(-n-1)+2780446/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 1)^(-n)+2780446/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 2)^(-n)+2780446/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 3)^(-n)+2780446/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 4)^(-n)+2780446/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 5)^(-n)+2780446/24254501*RootOf(_Z^6+2*_Z^5+4*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{11423 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{24254501}-\frac{11423 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{24254501}-\frac{11423 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{24254501}-\frac{11423 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{24254501}-\frac{11423 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{24254501}-\frac{11423 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{24254501}+\frac{113561 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{24254501}+\frac{113561 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{24254501}+\frac{113561 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{24254501}+\frac{113561 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{24254501}+\frac{113561 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{24254501}+\frac{113561 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{24254501}-\frac{282792 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{24254501}-\frac{282792 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{24254501}-\frac{282792 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{24254501}-\frac{282792 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{24254501}-\frac{282792 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{24254501}-\frac{282792 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{24254501}+\frac{305838 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{24254501}+\frac{305838 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{24254501}+\frac{305838 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{24254501}+\frac{305838 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{24254501}+\frac{305838 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{24254501}+\frac{305838 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{24254501}+\frac{1416856 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{24254501}+\frac{1416856 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{24254501}+\frac{1416856 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{24254501}+\frac{1416856 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{24254501}+\frac{1416856 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{24254501}+\frac{1416856 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{24254501}+\frac{2780446 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{24254501}+\frac{2780446 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{24254501}+\frac{2780446 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{24254501}+\frac{2780446 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{24254501}+\frac{2780446 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{24254501}+\frac{2780446 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{24254501}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(n+6) = a(n)+2*a(n+1)+4*a(n+2)+a(n+3)-4*a(n+4)+4*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) = 20
a \! \left(5\right) = 64
a \! \left(n +6\right) = a \! \left(n \right)+2 a \! \left(n +1\right)+4 a \! \left(n +2\right)+a \! \left(n +3\right)-4 a \! \left(n +4\right)+4 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15771/
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[13,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[6,x]
F[10,x] = F[11,x]+F[4,x]
F[11,x] = F[12,x]
F[12,x] = F[4,x]*F[7,x]
F[13,x] = F[14,x]+F[2,x]
F[14,x] = F[15,x]+F[16,x]+F[54,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[24,x]
F[18,x] = F[19,x]+F[7,x]
F[19,x] = F[15,x]+F[20,x]+F[22,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[18,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[10,x]
F[24,x] = F[25,x]+F[34,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[2,x]+F[25,x]
F[29,x] = F[30,x]+F[32,x]
F[30,x] = F[31,x]
F[31,x] = F[2,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[25,x]*F[4,x]
F[34,x] = 2*F[15,x]+F[35,x]+F[37,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[24,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]+F[53,x]
F[40,x] = F[15,x]+F[31,x]+F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[48,x]
F[43,x] = F[4,x]+F[44,x]
F[44,x] = F[15,x]+F[45,x]+F[47,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[43,x]
F[47,x] = x^2
F[48,x] = F[30,x]+F[49,x]
F[49,x] = 2*F[15,x]+F[50,x]+F[52,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[48,x]
F[52,x] = F[4,x]*F[40,x]
F[53,x] = F[33,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[13,x]+F[56,x]
F[56,x] = F[40,x]+F[57,x]
F[57,x] = 2*F[15,x]+F[33,x]+F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[64,x]
F[60,x] = F[11,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[60,x]
F[64,x] = F[32,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[64,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_{13}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{2}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{54}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{24}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{2}\! \left(x \right)+F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{24}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{53}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{48}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{43}\! \left(x \right)
F_{47}\! \left(x \right) = x^{2}
F_{48}\! \left(x \right) = F_{30}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{48}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{53}\! \left(x \right) = F_{33}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{40}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{33}\! \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_{60}\! \left(x \right)+F_{64}\! \left(x \right)
F_{60}\! \left(x \right) = F_{11}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{60}\! \left(x \right)
F_{64}\! \left(x \right) = F_{32}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{64}\! \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_13(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_6(x))
Eq(F_10(x), F_11(x) + F_4(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_4(x)*F_7(x))
Eq(F_13(x), F_14(x) + F_2(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_54(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_24(x))
Eq(F_18(x), F_19(x) + F_7(x))
Eq(F_19(x), F_15(x) + F_20(x) + F_22(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_18(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_10(x))
Eq(F_24(x), F_25(x) + F_34(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_2(x) + F_25(x))
Eq(F_29(x), F_30(x) + F_32(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_2(x)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_25(x)*F_4(x))
Eq(F_34(x), 2*F_15(x) + F_35(x) + F_37(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_24(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x) + F_53(x))
Eq(F_40(x), F_15(x) + F_31(x) + F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_48(x))
Eq(F_43(x), F_4(x) + F_44(x))
Eq(F_44(x), F_15(x) + F_45(x) + F_47(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_43(x))
Eq(F_47(x), x**2)
Eq(F_48(x), F_30(x) + F_49(x))
Eq(F_49(x), 2*F_15(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_48(x))
Eq(F_52(x), F_4(x)*F_40(x))
Eq(F_53(x), F_33(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_13(x) + F_56(x))
Eq(F_56(x), F_40(x) + F_57(x))
Eq(F_57(x), 2*F_15(x) + F_33(x) + F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_64(x))
Eq(F_60(x), F_11(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_60(x))
Eq(F_64(x), F_32(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_64(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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": [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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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": [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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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": [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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "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": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": 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