0123_0132_0321_1203_2031
Counting sequence:
1, 1, 2, 6, 19, 54, 147, 400, 1097, 3019, 8309, 22855, 62849, 172826, 475266, 1306992, 3594264, 9884301, 27181996, 74750936, 205566332, 565310927, 1554614743, 4275217170, 11756920366, 32331732113, 88912816386, 244511766129, 672411539904, 1849143238175, 5085175539505, 13984319728983, 38457118493381, 105757733767415, 290835577126050, 799802812608206, 2199471417414312, 6048584025662940, 16633709547594023, 45742985819461592, 125793993558227584, 345935634323961352, 951328912534507457, 2616170784466078225, 7194514413799643978, 19785037719139003998, 54409192203010344283, 149626209371358692758, 411474635523131100171, 1131562287050112259728, 3111815647752365371577, 8557546267152711584803, 23533398634123390275933, 64717248844840814862623, 177973541482953897314817, 489430283786078665623010, 1345941653410681591690882, 3701362572769771129450816, 10178810396709380406153824, 27991902726413481317571493, 76978211373138505395565428, 211691398191952258719717304, 582154966568906802705344948, 1600936117364304492437614351, 4402601710996697166716200455, 12107230023382830030545987050, 33295089690481558789052167894, 91562066249359362124191090281, 251797248596954226409391118210, 692446741299265674331790438905, 1904240384705188744194630521344, 5236693635005500841289535473583, 14400997083754581594616949226105, 39602988347453729517634284200751, 108908895469316250121006954857741, 299501325715211211272160281840983, 823633769478832047725997324591002, 2265006956499946972535717209559630, 6228807879307095099640802498004120, 17129328228321076999052017802992436, 47105945670329011204239405841996175, 129542156465144682606463631466706352, 356243146440219173786968810763690752, 979674747191462087852187638928354896, 2694122314703146189177124935508358721, 7408882455518594856096353836372130145, 20374553501198221409292337826602119570, 56030370688899212721272003732054629110, 154084477942097171220730110879506840099, 423734950363132636842288161514718510198, 1165278362605215776980815403154225059779, 3204535432331511340951994393238668046928, 8812527260962411801037900002052847390313, 24234600729224086507067012046828817984507, 66645566602283411910967055106256775192965, 183276448304894387684718957461573068748023, 504013368266656086086217611675806793878657, 1386045385214484455389105801661557796247274, 3811648521310588190410100915568208963233602, 10482098641929352303121601915072472006142544, 28825950589315675278811859010462393711699368
Generating function in Maple syntax:
(x-1)^3/(x^2-x+1)/(x^5+2*x^4-x^2+3*x-1)
Generating function in latex syntax:
\frac{\left(x -1\right)^{3}}{\left(x^{2}-x +1\right) \left(x^{5}+2 x^{4}-x^{2}+3 x -1\right)}
Generating function in sympy syntax:
(x - 1)**3/((x**2 - x + 1)*(x**5 + 2*x**4 - x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-x+1)*(x^5+2*x^4-x^2+3*x-1)*F(x)-(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-x +1\right) \left(x^{5}+2 x^{4}-x^{2}+3 x -1\right) F \! \left(x \right)-\left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
36/377*I*(I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)^(-n+4)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)+59/18*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)^(-n+3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)-29/18*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)^(-n+1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)+I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)^(-n+4)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)+119/36*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)^(-n+2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)-29/18*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)^(-n+1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)-47/36*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)^(-n)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)+119/36*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)^(-n+2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)+119/36*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)^(-n+2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)-29/18*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)^(-n+1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)-29/18*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)^(-n+1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)+59/18*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)^(-n+3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)+59/18*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)^(-n+3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)+I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)^(-n+4)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)+119/36*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)^(-n+2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)+I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)^(-n+4)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)+59/18*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)^(-n+3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)+59/18*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)^(-n+3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)+119/36*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)^(-n+2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)-29/18*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)^(-n+1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)+(-47/36*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)^(-n)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)+(-47/36*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)^(-n)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)+RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)*(-47/36*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)^(-n)*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)+(-47/36*I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)^(-n)-377/108*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)*3^(1/2)*(-(1/2+1/2*I*3^(1/2))^(-n)+(1/2-1/2*I*3^(1/2))^(-n))+I*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)^(-n+4))*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)))*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2))*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1))*(RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)^4+2*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)^3-RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 2)+3)*(RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)^4+2*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)^3-RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 4)+3)*(RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)^4+2*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)^3-RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 5)+3)*(RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)^4+2*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)^3-RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 1)+3)*(RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)^4+2*RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)^3-RootOf(_Z^5+2*_Z^4-_Z^2+3*_Z-1,index = 3)+3)
Explicit closed form in latex syntax:
\frac{36 \,\mathrm{I}}{377} \left(\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)+\frac{59 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)}{18}-\frac{29 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{18}+\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)+\frac{119 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{36}-\frac{29 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{18}-\frac{47 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{36}+\frac{119 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)}{36}+\frac{119 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{36}-\frac{29 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{18}-\frac{29 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)}{18}+\frac{59 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{18}+\frac{59 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{18}+\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)+\frac{119 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{36}+\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)+\frac{59 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{18}+\frac{59 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{18}+\frac{119 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{36}-\frac{29 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{18}+\left(-\frac{47 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{36}+\left(-\frac{47 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)}{36}+\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right) \left(-\frac{47 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)}{36}+\left(-\frac{47 \,\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{36}-\frac{377 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right) \sqrt{3}\, \left(-\left(\frac{1}{2}+\frac{\mathrm{I} \sqrt{3}}{2}\right)^{-n}+\left(\frac{1}{2}-\frac{\mathrm{I} \sqrt{3}}{2}\right)^{-n}\right)}{108}+\mathrm{I} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)\right)\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)\right) \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{4}+2 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{3}-\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)+3\right) \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{4}+2 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{3}-\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)+3\right) \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{4}+2 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{3}-\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)+3\right) \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{4}+2 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{3}-\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)+3\right) \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{4}+2 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{3}-\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)+3\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(6) = 147
a(n+7) = a(n)+a(n+1)-a(n+2)+a(n+3)+4*a(n+4)-5*a(n+5)+4*a(n+6), n >= 7
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) = 54
a \! \left(6\right) = 147
a \! \left(n +7\right) = a \! \left(n \right)+a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+4 a \! \left(n +4\right)-5 a \! \left(n +5\right)+4 a \! \left(n +6\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16235/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[4,x]
F[18,x] = F[12,x]*F[4,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,x]+F[21,x]+F[41,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[28,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[16,x]
F[28,x] = F[20,x]+F[29,x]
F[29,x] = 2*F[15,x]+F[30,x]+F[34,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[24,x]
F[33,x] = F[29,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[2,x]*F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[36,x]*F[4,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[2,x]+F[37,x]
F[44,x] = F[45,x]+F[57,x]
F[45,x] = F[15,x]+F[46,x]+F[56,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[50,x]
F[48,x] = F[12,x]+F[49,x]
F[49,x] = F[25,x]
F[50,x] = F[45,x]+F[51,x]
F[51,x] = 2*F[15,x]+F[34,x]+F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[49,x]
F[55,x] = F[51,x]
F[56,x] = F[4,x]*F[43,x]
F[57,x] = 2*F[15,x]+F[40,x]+F[58,x]
F[58,x] = F[4,x]*F[45,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{41}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{16}\! \left(x \right)
F_{28}\! \left(x \right) = F_{20}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \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_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{24}\! \left(x \right)
F_{33}\! \left(x \right) = F_{29}\! \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_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{2}\! \left(x \right)+F_{37}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{57}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{56}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{25}\! \left(x \right)
F_{50}\! \left(x \right) = F_{45}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{34}\! \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_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{49}\! \left(x \right)
F_{55}\! \left(x \right) = F_{51}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_12(x)*F_4(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_21(x) + F_41(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_28(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_16(x))
Eq(F_28(x), F_20(x) + F_29(x))
Eq(F_29(x), 2*F_15(x) + F_30(x) + F_34(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_24(x))
Eq(F_33(x), F_29(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_2(x)*F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_36(x)*F_4(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_2(x) + F_37(x))
Eq(F_44(x), F_45(x) + F_57(x))
Eq(F_45(x), F_15(x) + F_46(x) + F_56(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_50(x))
Eq(F_48(x), F_12(x) + F_49(x))
Eq(F_49(x), F_25(x))
Eq(F_50(x), F_45(x) + F_51(x))
Eq(F_51(x), 2*F_15(x) + F_34(x) + F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_49(x))
Eq(F_55(x), F_51(x))
Eq(F_56(x), F_4(x)*F_43(x))
Eq(F_57(x), 2*F_15(x) + F_40(x) + F_58(x))
Eq(F_58(x), F_4(x)*F_45(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [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, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [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, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 3, 1], "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], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 3, 1], "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], "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": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[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], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": 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