0123_0231_0312_1032_2103
Counting sequence:
1, 1, 2, 6, 19, 52, 138, 378, 1047, 2889, 7943, 21845, 60132, 165541, 455646, 1254075, 3451697, 9500586, 26149704, 71974877, 198104784, 545267342, 1500804540, 4130843043, 11369809985, 31294479407, 86135517795, 237081030378, 652546313267, 1796080815692, 4943566817007, 13606766841515, 37451522492822, 103082279070010, 283725615172415, 780931751165444, 2149451326805974, 5916190498559602, 16283834659871959, 44819934600649857, 123363235967918899, 339547304655224897, 934576425415613384, 2572345835080317801, 7080173344103767142, 19487603066013465975, 53638047375599380801, 147634376404382453750, 406351650791563648556, 1118449971629558502997, 3078442862484689603888, 8473164377460182181866, 23321697940988585451644, 64191082648835017180323, 176680750349127628319857, 486299439981439427359099, 1338499779172062336210847, 3684112116008274389264110, 10140219889848943421855343, 27910133073229517040902044, 76820378317948324965275283, 211441862689401001062090003, 581976583251452203515170018, 1601843358477055326810924054, 4408943966029666355281026519, 12135260787341113381925304652, 33401321384765352129492663994, 91934428917437912197229362626, 253042061510490148212216411511, 696477757543709087258275897817, 1916998556909891525758232086703, 5276383096791114294317906473325, 14522816662407961450988241498524, 39972875346784942373727159032957, 110022098373352215505714576244478, 302826905131536857929568318063075, 833506503033175432011097696644337, 2294159068517463500080541476402354, 6314486824647401650889447117280784, 17380113003416226611288553219613725, 47837351854540480473170754493202240, 131668432305664430613173092671135046, 362406684185755909784068835891625884, 997495014124687663168181686888600339, 2745524149034771953909677061942295873, 7556832611887987722819490430762668487, 20799569052841933254210784772505020795, 57249127379553564940781075501118152130, 157573581327279279783773820285744701787, 433708506466641622986083085695384723644, 1193747498769074092835074414759921347591, 3285693200778910582267080533717979241467, 9043604129664580679817563735008046186430, 24891787107420066308088773488669795153554, 68512625775903383116613936957066282285631, 188575447413727008999627101113805557163684, 519038629224372764930913280258366391032350, 1428611743055078862752384633780385963643218, 3932137990277802610858858608502194130249831, 10822890998726617615331264141637842636141393, 29789129949135417911760077131974595657950187
Generating function in Maple syntax:
-(x^3+x^2+x-1)/(2*x^8+4*x^7+4*x^6-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}{2 x^{8}+4 x^{7}+4 x^{6}-x^{5}-3 x^{4}-2 x^{3}-x^{2}-2 x +1}
Generating function in sympy syntax:
(-x**3 - x**2 - x + 1)/(2*x**8 + 4*x**7 + 4*x**6 - x**5 - 3*x**4 - 2*x**3 - x**2 - 2*x + 1)
Implicit equation for the generating function in Maple syntax:
(2*x^8+4*x^7+4*x^6-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(2 x^{8}+4 x^{7}+4 x^{6}-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:
-1344869652/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 1)^(-n+6)-1344869652/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 2)^(-n+6)-1344869652/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 3)^(-n+6)-1344869652/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 4)^(-n+6)-1344869652/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 5)^(-n+6)-1344869652/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 6)^(-n+6)-1344869652/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 7)^(-n+6)-1344869652/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 8)^(-n+6)-3051180216/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 1)^(-n+5)-3051180216/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 2)^(-n+5)-3051180216/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 3)^(-n+5)-3051180216/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 4)^(-n+5)-3051180216/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 5)^(-n+5)-3051180216/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 6)^(-n+5)-3051180216/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 7)^(-n+5)-3051180216/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 8)^(-n+5)-5070936826/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 1)^(-n+4)-5070936826/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 2)^(-n+4)-5070936826/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 3)^(-n+4)-5070936826/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 4)^(-n+4)-5070936826/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 5)^(-n+4)-5070936826/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 6)^(-n+4)-5070936826/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 7)^(-n+4)-5070936826/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 8)^(-n+4)-5217027252/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 1)^(-n+3)-5217027252/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 2)^(-n+3)-5217027252/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 3)^(-n+3)-5217027252/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 4)^(-n+3)-5217027252/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 5)^(-n+3)-5217027252/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 6)^(-n+3)-5217027252/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 7)^(-n+3)-5217027252/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 8)^(-n+3)-899190950/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 1)^(-n+2)-899190950/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 2)^(-n+2)-899190950/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 3)^(-n+2)-899190950/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 4)^(-n+2)-899190950/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 5)^(-n+2)-899190950/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 6)^(-n+2)-899190950/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 7)^(-n+2)-899190950/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 8)^(-n+2)+4059297833/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 1)^(-n+1)+4059297833/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 2)^(-n+1)+4059297833/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 3)^(-n+1)+4059297833/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 4)^(-n+1)+4059297833/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 5)^(-n+1)+4059297833/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 6)^(-n+1)+4059297833/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 7)^(-n+1)+4059297833/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 8)^(-n+1)+2381508199/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 1)^(-n-1)+2381508199/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 2)^(-n-1)+2381508199/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 3)^(-n-1)+2381508199/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 4)^(-n-1)+2381508199/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 5)^(-n-1)+2381508199/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 6)^(-n-1)+2381508199/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 7)^(-n-1)+2381508199/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 8)^(-n-1)+9805626918/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 1)^(-n)+9805626918/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 2)^(-n)+9805626918/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 3)^(-n)+9805626918/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 4)^(-n)+9805626918/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 5)^(-n)+9805626918/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 6)^(-n)+9805626918/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 7)^(-n)+9805626918/54555387391*RootOf(2*_Z^8+4*_Z^7+4*_Z^6-_Z^5-3*_Z^4-2*_Z^3-_Z^2-2*_Z+1,index = 8)^(-n)
Explicit closed form in latex syntax:
-\frac{1344869652 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +6}}{54555387391}-\frac{1344869652 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +6}}{54555387391}-\frac{1344869652 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +6}}{54555387391}-\frac{1344869652 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +6}}{54555387391}-\frac{1344869652 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +6}}{54555387391}-\frac{1344869652 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +6}}{54555387391}-\frac{1344869652 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +6}}{54555387391}-\frac{1344869652 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +6}}{54555387391}-\frac{3051180216 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{54555387391}-\frac{3051180216 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{54555387391}-\frac{3051180216 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{54555387391}-\frac{3051180216 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{54555387391}-\frac{3051180216 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{54555387391}-\frac{3051180216 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{54555387391}-\frac{3051180216 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{54555387391}-\frac{3051180216 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +5}}{54555387391}-\frac{5070936826 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{54555387391}-\frac{5070936826 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{54555387391}-\frac{5070936826 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{54555387391}-\frac{5070936826 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{54555387391}-\frac{5070936826 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{54555387391}-\frac{5070936826 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{54555387391}-\frac{5070936826 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{54555387391}-\frac{5070936826 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +4}}{54555387391}-\frac{5217027252 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{54555387391}-\frac{5217027252 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{54555387391}-\frac{5217027252 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{54555387391}-\frac{5217027252 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{54555387391}-\frac{5217027252 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{54555387391}-\frac{5217027252 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{54555387391}-\frac{5217027252 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{54555387391}-\frac{5217027252 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +3}}{54555387391}-\frac{899190950 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{54555387391}-\frac{899190950 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{54555387391}-\frac{899190950 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{54555387391}-\frac{899190950 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{54555387391}-\frac{899190950 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{54555387391}-\frac{899190950 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{54555387391}-\frac{899190950 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{54555387391}-\frac{899190950 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +2}}{54555387391}+\frac{4059297833 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{54555387391}+\frac{4059297833 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{54555387391}+\frac{4059297833 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{54555387391}+\frac{4059297833 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{54555387391}+\frac{4059297833 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{54555387391}+\frac{4059297833 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{54555387391}+\frac{4059297833 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{54555387391}+\frac{4059297833 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +1}}{54555387391}+\frac{2381508199 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{54555387391}+\frac{2381508199 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{54555387391}+\frac{2381508199 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{54555387391}+\frac{2381508199 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{54555387391}+\frac{2381508199 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{54555387391}+\frac{2381508199 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{54555387391}+\frac{2381508199 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{54555387391}+\frac{2381508199 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n -1}}{54555387391}+\frac{9805626918 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{54555387391}+\frac{9805626918 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{54555387391}+\frac{9805626918 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{54555387391}+\frac{9805626918 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{54555387391}+\frac{9805626918 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{54555387391}+\frac{9805626918 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{54555387391}+\frac{9805626918 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{54555387391}+\frac{9805626918 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1, \mathit{index} =8\right)^{-n}}{54555387391}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 52
a(6) = 138
a(7) = 378
a(n+8) = -2*a(n)-4*a(n+1)-4*a(n+2)+a(n+3)+3*a(n+4)+2*a(n+5)+a(n+6)+2*a(n+7), n >= 8
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) = 52
a \! \left(6\right) = 138
a \! \left(7\right) = 378
a \! \left(n +8\right) = -2 a \! \left(n \right)-4 a \! \left(n +1\right)-4 a \! \left(n +2\right)+a \! \left(n +3\right)+3 a \! \left(n +4\right)+2 a \! \left(n +5\right)+a \! \left(n +6\right)+2 a \! \left(n +7\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17007/
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[11,x]+F[14,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[61,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[35,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[19,x]+F[27,x]+F[32,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]+F[4,x]
F[30,x] = F[31,x]
F[31,x] = F[11,x]*F[4,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[11,x]+F[34,x]
F[34,x] = F[15,x]
F[35,x] = F[36,x]+F[41,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[2,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[2,x]*F[4,x]
F[41,x] = 2*F[19,x]+F[42,x]+F[56,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[39,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[19,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[4,x]+F[51,x]
F[51,x] = F[19,x]+F[27,x]+F[31,x]
F[52,x] = F[39,x]+F[53,x]
F[53,x] = 2*F[19,x]+F[42,x]+F[46,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[2,x]+F[47,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[47,x]+F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[39,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[55,x]+F[63,x]
F[63,x] = F[47,x]+F[64,x]
F[64,x] = 2*F[19,x]+F[59,x]+F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[69,x]
F[67,x] = F[68,x]
F[68,x] = x^2
F[69,x] = F[70,x]
F[70,x] = F[39,x]*F[4,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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{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_{61}\! \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_{35}\! \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_{19}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \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_{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) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{11}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{56}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{39}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{19}\! \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_{4}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{19}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{52}\! \left(x \right) = F_{39}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{2}\! \left(x \right)+F_{47}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{47}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{39}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{55}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{47}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{59}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{69}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = x^{2}
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \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_11(x) + F_14(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_61(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_35(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_19(x) + F_27(x) + F_32(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x) + F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_11(x)*F_4(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_11(x) + F_34(x))
Eq(F_34(x), F_15(x))
Eq(F_35(x), F_36(x) + F_41(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_2(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_2(x)*F_4(x))
Eq(F_41(x), 2*F_19(x) + F_42(x) + F_56(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_39(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_19(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_4(x) + F_51(x))
Eq(F_51(x), F_19(x) + F_27(x) + F_31(x))
Eq(F_52(x), F_39(x) + F_53(x))
Eq(F_53(x), 2*F_19(x) + F_42(x) + F_46(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_2(x) + F_47(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_47(x) + F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_39(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_55(x) + F_63(x))
Eq(F_63(x), F_47(x) + F_64(x))
Eq(F_64(x), 2*F_19(x) + F_59(x) + F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_69(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), x**2)
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_39(x)*F_4(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": [0, 3, 1, 2], "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, 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": [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, 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": [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, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "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": [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": [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, 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, 2]]}, {"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": [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": [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": [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, 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", 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