0123_0231_1032_2031_2103
Counting sequence:
1, 1, 2, 6, 19, 51, 131, 344, 921, 2465, 6564, 17454, 46452, 123706, 329428, 877109, 2335221, 6217513, 16554441, 44076932, 117356066, 312463425, 831942596, 2215071912, 5897694219, 15702781302, 41809107589, 111317958316, 296387290523, 789139747890, 2101107426502, 5594259350829, 14894877500927, 39658042648958, 105590686887771, 281138261245346, 748538760983862, 1993006125331554, 5306436517031058, 14128540876597099, 37617649181210758, 100158080178265646, 266673788590347378, 710026683764361043, 1890466604618923782, 5033422073985733097, 13401632016659639543, 35682233293769770104, 95004979337463560345, 252953508391744334379, 673496040459219684405, 1793202709059761442819, 4774454135746294899793, 12712122382581800855239, 33846393928020786810260, 90117003868719668829561, 239939132173000938150796, 638844886940544396636891, 1700943009478804915943951, 4528810014196884249682847, 12058087796236394069246988, 32105007904052936123193488, 85480513157402658143496348, 227594341402746193759063743, 605976524066572630278699675, 1613430041610756349655614864, 4295804203276534820165621296, 11437703077888020008753534903, 30453215627972072124174130779, 81082568394056196220275646297, 215884686126152759910548296488, 574799227835066800395881226977, 1530419587643758935350701616687, 4074786465989757221609338262292, 10849236952708317406305569119844, 28886407530417947091456253806468, 76910896466786778150721975030896, 204777488827431521010945987662652, 545225993414048336976263188019653, 1451679994693411291854795392832715, 3865140019824234313588155879950519, 10291047219399066868798062964376025, 27400211202883481274694589520257878, 72953855711339532731241131394738462, 194241753238415012693863750131450869, 517174292342012490169449404265736280, 1376991528341313653224800525040231116, 3666279815528482054773461008551704009, 9761576167387865862285336313585478622, 25990479195865543708841931305004509589, 69200403423321376968313131725669485740, 184248077838910895232752338290519539917, 490565842220114821058172361437393168190, 1306145759433845709855105667775355742352, 3477650904445839999248289084590738439519, 9259346229807603985172102592937174794655, 24653277444796935567703392559476749468544, 65640065041909790144536942965335530214309, 174768573807434266735379838187486907703164, 465326388253620217298885326777188591820056, 1238944982429949061433659975888165355653916
Generating function in Maple syntax:
(x-1)^2/(x^6+x^5-3*x^4-2*x^3+2*x^2-3*x+1)
Generating function in latex syntax:
\frac{\left(x -1\right)^{2}}{x^{6}+x^{5}-3 x^{4}-2 x^{3}+2 x^{2}-3 x +1}
Generating function in sympy syntax:
(x - 1)**2/(x**6 + x**5 - 3*x**4 - 2*x**3 + 2*x**2 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^6+x^5-3*x^4-2*x^3+2*x^2-3*x+1)*F(x)-(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{6}+x^{5}-3 x^{4}-2 x^{3}+2 x^{2}-3 x +1\right) F \! \left(x \right)-\left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
-1454/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+4)-1454/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+4)-1454/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+4)-1454/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+4)-1454/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+4)-1454/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+4)-2773/42341*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+3)-2773/42341*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+3)-2773/42341*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+3)-2773/42341*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+3)-2773/42341*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+3)-2773/42341*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+3)-121/9771*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+2)-121/9771*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+2)-121/9771*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+2)-121/9771*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+2)-121/9771*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+2)-121/9771*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+2)+12866/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+1)+12866/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+1)+12866/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+1)+12866/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+1)+12866/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+1)+12866/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+1)+2948/42341*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n-1)+2948/42341*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n-1)+2948/42341*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n-1)+2948/42341*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n-1)+2948/42341*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n-1)+2948/42341*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n-1)+18817/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n)+18817/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n)+18817/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n)+18817/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n)+18817/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n)+18817/127023*RootOf(_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{1454 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{127023}-\frac{1454 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{127023}-\frac{1454 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{127023}-\frac{1454 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{127023}-\frac{1454 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{127023}-\frac{1454 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{127023}-\frac{2773 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{42341}-\frac{2773 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{42341}-\frac{2773 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{42341}-\frac{2773 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{42341}-\frac{2773 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{42341}-\frac{2773 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{42341}-\frac{121 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{9771}-\frac{121 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{9771}-\frac{121 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{9771}-\frac{121 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{9771}-\frac{121 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{9771}-\frac{121 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{9771}+\frac{12866 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{127023}+\frac{12866 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{127023}+\frac{12866 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{127023}+\frac{12866 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{127023}+\frac{12866 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{127023}+\frac{12866 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{127023}+\frac{2948 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{42341}+\frac{2948 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{42341}+\frac{2948 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{42341}+\frac{2948 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{42341}+\frac{2948 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{42341}+\frac{2948 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{42341}+\frac{18817 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{127023}+\frac{18817 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{127023}+\frac{18817 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{127023}+\frac{18817 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{127023}+\frac{18817 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{127023}+\frac{18817 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{127023}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(n+6) = -a(n)-a(n+1)+3*a(n+2)+2*a(n+3)-2*a(n+4)+3*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) = 51
a \! \left(n +6\right) = -a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+2 a \! \left(n +3\right)-2 a \! \left(n +4\right)+3 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17258/
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[17,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[16,x]*F[4,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[36,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[27,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[1,x]+F[4,x]
F[26,x] = F[15,x]
F[27,x] = F[28,x]+F[33,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[2,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[2,x]*F[4,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[31,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[47,x]
F[38,x] = F[2,x]+F[39,x]
F[39,x] = F[19,x]+F[40,x]+F[46,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[44,x]
F[42,x] = F[4,x]+F[43,x]
F[43,x] = F[15,x]
F[44,x] = F[31,x]+F[45,x]
F[45,x] = F[34,x]
F[46,x] = F[38,x]*F[4,x]
F[47,x] = F[18,x]+F[48,x]
F[48,x] = 2*F[19,x]+F[34,x]+F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[4,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[31,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_{17}\! \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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{36}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{33}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{2}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{47}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{19}\! \left(x \right)+F_{40}\! \left(x \right)+F_{46}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right)
F_{44}\! \left(x \right) = F_{31}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{34}\! \left(x \right)
F_{46}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{47}\! \left(x \right) = F_{18}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{34}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{31}\! \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_17(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_16(x)*F_4(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_36(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_27(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_1(x) + F_4(x))
Eq(F_26(x), F_15(x))
Eq(F_27(x), F_28(x) + F_33(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_2(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_2(x)*F_4(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_47(x))
Eq(F_38(x), F_2(x) + F_39(x))
Eq(F_39(x), F_19(x) + F_40(x) + F_46(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_44(x))
Eq(F_42(x), F_4(x) + F_43(x))
Eq(F_43(x), F_15(x))
Eq(F_44(x), F_31(x) + F_45(x))
Eq(F_45(x), F_34(x))
Eq(F_46(x), F_38(x)*F_4(x))
Eq(F_47(x), F_18(x) + F_48(x))
Eq(F_48(x), 2*F_19(x) + F_34(x) + F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_54(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_4(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_31(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, 3, 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, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 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, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 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, 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, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 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, 3, 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": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"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]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [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": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 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, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"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]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 2], [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": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[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": []}, {"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, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 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, 3, 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": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"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]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 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, 3, 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": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"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]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[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", 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