0123_0321_1032
Counting sequence:
1, 1, 2, 6, 21, 72, 246, 845, 2901, 9955, 34165, 117254, 402409, 1381046, 4739681, 16266344, 55825262, 191589456, 657525254, 2256593172, 7744512803, 26578773396, 91216996231, 313052083987, 1074378803719, 3687213319833, 12654328267540, 43429009935872, 149046149596743, 511518792217847, 1755506435422361, 6024808651598556, 20676836356708481, 70961848988957543, 243537450558670629, 835808123233180065, 2868450898455459081, 9844377349458253459, 33785401538757258453, 115949776874194762310, 397933727138117041327, 1365688278691938541351, 4686972597095795317234, 16085451174090520267484, 55204448951627136910663, 189458856395746418072026, 650213143115272009606761, 2231498381879460460711595, 7658388762295216304566036, 26283200073419466154225254, 90202603646927234918164348, 309570740319144244602514703, 1062431009606535242696560121, 3646209099121884492540050828, 12513603871034305185214630040, 42946050976307513736515173429, 147388659051995282593613566761, 505830369109599623632069201894, 1735984057113178670772395913257, 5957808843814512801949990882667, 20446896429717769717234132905724, 70172706873879926207893228461167, 240829155022792098446739463679598, 826513390928925664416083181284140, 2836551850710019439885995805781112, 9734901442701898176515144551415705, 33409685804050425556128147254244133, 114660339613625093657733882642113481, 393508444150587229802712637988491499, 1350500932926027434324589773403843165, 4634850400150806341463608510665653370, 15906570449555357316723379691838625476, 54590539418151473160555123904164270986, 187351950152652450336402528044166768124, 642982348225906917350881715656902934755, 2206682662194046599716098237032092170179, 7573222476578721620470491177529579581613, 25990913719660927535990564537188810024264, 89199491771438744890664375523055127188346, 306128111466285455855354651160672686686704, 1050616082769222758676210152908590205248073, 3605660872131011526025919785170524162306672, 12374444421742501344258210460095825800080679, 42468462835855476524563413500306172548775682, 145749600884826545427895380667346372816298902, 500205204982157605412809865511141556044154051, 1716678780403372310059699238760186298975859845, 5891554117659229882863578849104070314623029774, 20219513584918568660393880853172294159530570036, 69392340534612294311192736263923239460654862980, 238150977502409063466179197033046477861086279212, 817322022119481003988435516227721492678236404022, 2805007541212884395426086267646052398197971631324, 9626642979541484056420389374202740365987737313021, 33038148273741855013430010949115023128670068836049, 113385241737690458683079566660696242111474749833598, 389132373200600855133872339829032426021515326069731, 1335482480365844255727164747117435541639714479453778, 4583307836083563566011089835963287545559826665984537, 15729678995527058718783515832697400557886266019821549, 53983456959711420259068865322123949397859721414602481
Generating function in Maple syntax:
(x^7+x^6-x^5+3*x^3+2*x^2+2*x-1)/(x^7+x^6-x^5-x^4+2*x^3+x^2+3*x-1)
Generating function in latex syntax:
\frac{x^{7}+x^{6}-x^{5}+3 x^{3}+2 x^{2}+2 x -1}{x^{7}+x^{6}-x^{5}-x^{4}+2 x^{3}+x^{2}+3 x -1}
Generating function in sympy syntax:
(x**7 + x**6 - x**5 + 3*x**3 + 2*x**2 + 2*x - 1)/(x**7 + x**6 - x**5 - x**4 + 2*x**3 + x**2 + 3*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^7+x^6-x^5-x^4+2*x^3+x^2+3*x-1)*F(x)-x^7-x^6+x^5-3*x^3-2*x^2-2*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}+x^{6}-x^{5}-x^{4}+2 x^{3}+x^{2}+3 x -1\right) F \! \left(x \right)-x^{7}-x^{6}+x^{5}-3 x^{3}-2 x^{2}-2 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,0)+922689/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 1)^(-n+5)+922689/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 2)^(-n+5)+922689/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 3)^(-n+5)+922689/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 4)^(-n+5)+922689/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 5)^(-n+5)+922689/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 6)^(-n+5)+922689/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 7)^(-n+5)+650133/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 1)^(-n+4)+650133/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 2)^(-n+4)+650133/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 3)^(-n+4)+650133/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 4)^(-n+4)+650133/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 5)^(-n+4)+650133/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 6)^(-n+4)+650133/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 7)^(-n+4)-155501/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 1)^(-n+3)-155501/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 2)^(-n+3)-155501/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 3)^(-n+3)-155501/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 4)^(-n+3)-155501/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 5)^(-n+3)-155501/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 6)^(-n+3)-155501/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 7)^(-n+3)+271720/9242623*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 1)^(-n+2)+271720/9242623*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 2)^(-n+2)+271720/9242623*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 3)^(-n+2)+271720/9242623*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 4)^(-n+2)+271720/9242623*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 5)^(-n+2)+271720/9242623*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 6)^(-n+2)+271720/9242623*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 7)^(-n+2)+5145758/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 1)^(-n+1)+5145758/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 2)^(-n+1)+5145758/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 3)^(-n+1)+5145758/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 4)^(-n+1)+5145758/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 5)^(-n+1)+5145758/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 6)^(-n+1)+5145758/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 7)^(-n+1)+1758049/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 1)^(-n-1)+1758049/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 2)^(-n-1)+1758049/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 3)^(-n-1)+1758049/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 4)^(-n-1)+1758049/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 5)^(-n-1)+1758049/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 6)^(-n-1)+1758049/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 7)^(-n-1)-685683/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 1)^(-n)-685683/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 2)^(-n)-685683/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 3)^(-n)-685683/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 4)^(-n)-685683/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 5)^(-n)-685683/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 6)^(-n)-685683/46213115*RootOf(_Z^7+_Z^6-_Z^5-_Z^4+2*_Z^3+_Z^2+3*_Z-1,index = 7)^(-n)
Explicit closed form in latex syntax:
\frac{922689 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{46213115}+\frac{922689 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{46213115}+\frac{922689 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{46213115}+\frac{922689 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{46213115}+\frac{922689 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{46213115}+\frac{922689 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{46213115}+\frac{922689 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{46213115}+\frac{650133 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{46213115}+\frac{650133 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{46213115}+\frac{650133 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{46213115}+\frac{650133 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{46213115}+\frac{650133 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{46213115}+\frac{650133 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{46213115}+\frac{650133 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{46213115}-\frac{155501 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{46213115}-\frac{155501 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{46213115}-\frac{155501 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{46213115}-\frac{155501 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{46213115}-\frac{155501 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{46213115}-\frac{155501 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{46213115}-\frac{155501 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{46213115}+\frac{271720 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{9242623}+\frac{271720 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{9242623}+\frac{271720 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{9242623}+\frac{271720 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{9242623}+\frac{271720 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{9242623}+\frac{271720 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{9242623}+\frac{271720 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{9242623}+\frac{5145758 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{46213115}+\frac{5145758 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{46213115}+\frac{5145758 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{46213115}+\frac{5145758 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{46213115}+\frac{5145758 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{46213115}+\frac{5145758 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{46213115}+\frac{5145758 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{46213115}+\frac{1758049 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{46213115}+\frac{1758049 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{46213115}+\frac{1758049 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{46213115}+\frac{1758049 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{46213115}+\frac{1758049 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{46213115}+\frac{1758049 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{46213115}+\frac{1758049 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{46213115}-\frac{685683 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{46213115}-\frac{685683 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{46213115}-\frac{685683 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{46213115}-\frac{685683 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{46213115}-\frac{685683 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{46213115}-\frac{685683 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{46213115}-\frac{685683 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{46213115}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 72
a(6) = 246
a(7) = 845
a(n+7) = a(n)+a(n+1)-a(n+2)-a(n+3)+2*a(n+4)+a(n+5)+3*a(n+6), 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) = 21
a \! \left(5\right) = 72
a \! \left(6\right) = 246
a \! \left(7\right) = 845
a \! \left(n +7\right) = a \! \left(n \right)+a \! \left(n +1\right)-a \! \left(n +2\right)-a \! \left(n +3\right)+2 a \! \left(n +4\right)+a \! \left(n +5\right)+3 a \! \left(n +6\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14650/
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[28,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]+F[19,x]+F[23,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[4,x]
F[21,x] = F[22,x]
F[22,x] = x^2
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[15,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[4,x]
F[28,x] = F[2,x]+F[29,x]
F[29,x] = F[18,x]+F[30,x]+F[94,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[45,x]
F[32,x] = F[15,x]+F[33,x]
F[33,x] = F[18,x]+F[34,x]+F[43,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[38,x]
F[36,x] = F[37,x]+F[4,x]
F[37,x] = F[22,x]
F[38,x] = F[39,x]+F[41,x]
F[39,x] = F[18,x]+F[19,x]+F[40,x]
F[40,x] = F[15,x]*F[4,x]
F[41,x] = F[42,x]
F[42,x] = F[25,x]*F[4,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[15,x]+F[39,x]
F[45,x] = F[46,x]+F[83,x]
F[46,x] = F[18,x]+F[30,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[2,x]+F[49,x]
F[49,x] = F[18,x]+F[50,x]+F[82,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[54,x]
F[52,x] = F[4,x]+F[53,x]
F[53,x] = F[18,x]+F[34,x]+F[40,x]
F[54,x] = F[49,x]+F[55,x]
F[55,x] = F[18,x]+F[56,x]+F[74,x]+F[75,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[62,x]
F[58,x] = F[25,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[39,x]
F[62,x] = F[63,x]+F[66,x]
F[63,x] = 2*F[18,x]+F[56,x]+F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[49,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[18,x]+F[56,x]+F[70,x]+F[74,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[49,x]+F[72,x]
F[72,x] = F[73,x]
F[73,x] = F[4,x]*F[49,x]
F[74,x] = F[4,x]*F[46,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[77,x]+F[79,x]
F[77,x] = F[49,x]+F[78,x]
F[78,x] = F[73,x]
F[79,x] = F[69,x]+F[80,x]
F[80,x] = F[81,x]
F[81,x] = F[4,x]*F[63,x]
F[82,x] = F[2,x]*F[4,x]
F[83,x] = F[18,x]+F[75,x]+F[84,x]+F[92,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[86,x]+F[89,x]
F[86,x] = F[87,x]+F[88,x]
F[87,x] = F[26,x]
F[88,x] = F[60,x]
F[89,x] = F[90,x]+F[91,x]
F[90,x] = 2*F[18,x]+F[64,x]+F[84,x]
F[91,x] = F[67,x]
F[92,x] = F[4,x]*F[93,x]
F[93,x] = F[46,x]+F[69,x]
F[94,x] = F[4,x]*F[95,x]
F[95,x] = F[96,x]+F[99,x]
F[96,x] = F[2,x]+F[97,x]
F[97,x] = F[18,x]+F[50,x]+F[98,x]
F[98,x] = F[4,x]*F[48,x]
F[99,x] = F[100,x]+F[46,x]
F[100,x] = F[101,x]+F[18,x]+F[56,x]+F[70,x]
F[101,x] = F[102,x]*F[4,x]
F[102,x] = F[46,x]+F[63,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_{28}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = x^{2}
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \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_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{2}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{94}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{45}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{43}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{22}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{41}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \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_{39}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{83}\! \left(x \right)
F_{46}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{82}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{40}\! \left(x \right)
F_{54}\! \left(x \right) = F_{49}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{25}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{39}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{66}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{49}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{49}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{79}\! \left(x \right)
F_{77}\! \left(x \right) = F_{49}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{73}\! \left(x \right)
F_{79}\! \left(x \right) = F_{69}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{82}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{83}\! \left(x \right) = F_{18}\! \left(x \right)+F_{75}\! \left(x \right)+F_{84}\! \left(x \right)+F_{92}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)+F_{89}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)+F_{88}\! \left(x \right)
F_{87}\! \left(x \right) = F_{26}\! \left(x \right)
F_{88}\! \left(x \right) = F_{60}\! \left(x \right)
F_{89}\! \left(x \right) = F_{90}\! \left(x \right)+F_{91}\! \left(x \right)
F_{90}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{84}\! \left(x \right)
F_{91}\! \left(x \right) = F_{67}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{46}\! \left(x \right)+F_{69}\! \left(x \right)
F_{94}\! \left(x \right) = F_{4}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{96}\! \left(x \right)+F_{99}\! \left(x \right)
F_{96}\! \left(x \right) = F_{2}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{46}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{70}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{4}\! \left(x \right)
F_{102}\! \left(x \right) = F_{46}\! \left(x \right)+F_{63}\! \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_28(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_23(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), x**2)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_15(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_4(x))
Eq(F_28(x), F_2(x) + F_29(x))
Eq(F_29(x), F_18(x) + F_30(x) + F_94(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_45(x))
Eq(F_32(x), F_15(x) + F_33(x))
Eq(F_33(x), F_18(x) + F_34(x) + F_43(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_38(x))
Eq(F_36(x), F_37(x) + F_4(x))
Eq(F_37(x), F_22(x))
Eq(F_38(x), F_39(x) + F_41(x))
Eq(F_39(x), F_18(x) + F_19(x) + F_40(x))
Eq(F_40(x), F_15(x)*F_4(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_25(x)*F_4(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_15(x) + F_39(x))
Eq(F_45(x), F_46(x) + F_83(x))
Eq(F_46(x), F_18(x) + F_30(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_2(x) + F_49(x))
Eq(F_49(x), F_18(x) + F_50(x) + F_82(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_54(x))
Eq(F_52(x), F_4(x) + F_53(x))
Eq(F_53(x), F_18(x) + F_34(x) + F_40(x))
Eq(F_54(x), F_49(x) + F_55(x))
Eq(F_55(x), F_18(x) + F_56(x) + F_74(x) + F_75(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_62(x))
Eq(F_58(x), F_25(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_39(x))
Eq(F_62(x), F_63(x) + F_66(x))
Eq(F_63(x), 2*F_18(x) + F_56(x) + F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_49(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_18(x) + F_56(x) + F_70(x) + F_74(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_49(x) + F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_4(x)*F_49(x))
Eq(F_74(x), F_4(x)*F_46(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_77(x) + F_79(x))
Eq(F_77(x), F_49(x) + F_78(x))
Eq(F_78(x), F_73(x))
Eq(F_79(x), F_69(x) + F_80(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_4(x)*F_63(x))
Eq(F_82(x), F_2(x)*F_4(x))
Eq(F_83(x), F_18(x) + F_75(x) + F_84(x) + F_92(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_86(x) + F_89(x))
Eq(F_86(x), F_87(x) + F_88(x))
Eq(F_87(x), F_26(x))
Eq(F_88(x), F_60(x))
Eq(F_89(x), F_90(x) + F_91(x))
Eq(F_90(x), 2*F_18(x) + F_64(x) + F_84(x))
Eq(F_91(x), F_67(x))
Eq(F_92(x), F_4(x)*F_93(x))
Eq(F_93(x), F_46(x) + F_69(x))
Eq(F_94(x), F_4(x)*F_95(x))
Eq(F_95(x), F_96(x) + F_99(x))
Eq(F_96(x), F_2(x) + F_97(x))
Eq(F_97(x), F_18(x) + F_50(x) + F_98(x))
Eq(F_98(x), F_4(x)*F_48(x))
Eq(F_99(x), F_100(x) + F_46(x))
Eq(F_100(x), F_101(x) + F_18(x) + F_56(x) + F_70(x))
Eq(F_101(x), F_102(x)*F_4(x))
Eq(F_102(x), F_46(x) + F_63(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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "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": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "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": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "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": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "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": 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"strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "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": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}], [{"patt": [0], "pos": [[0, 1]]}]]}], "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, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 1]]}], "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], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], 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