0231_0312_1302_2103
Counting sequence:
1, 1, 2, 6, 20, 64, 194, 571, 1669, 4895, 14422, 42590, 125820, 371549, 1096752, 3236866, 9552916, 28194574, 83216595, 245617802, 724952332, 2139721553, 6315443334, 18640177784, 55016928493, 162383837351, 479280057318, 1414607447290, 4175250243378, 12323357985699, 36372705942861, 107354970572032, 316860938543165, 935223158728741, 2760335059972966, 8147199492294959, 24046667561548362, 70974353974132159, 209482619937423376, 618293307355410464, 1824908500986322635, 5386264087652714490, 15897696134599722935, 46922456506924021643, 138492829778384037076, 408765127144624694649, 1206480721325922183751, 3560958688181877002441, 10510260590824553205333, 31021302789517985334701, 91560168127427997582372, 270242176623070868241033, 797626692038543778476457, 2354215569909784447421213, 6948527431348580159036742, 20508756326870281951008767, 60532118528790243566722689, 178662095116065764056374792, 527325740566643017467500098, 1556415402402573898858687974, 4593799844158794577789661924, 13558717663431975246449146919, 40018901761778078002557782619, 118116811484182097722264541236, 348624788312280012125180845002, 1028974973999008397793300480383, 3037045937673976482541250269381, 8963918715821840452019736992680, 26457235218971111278292056721241, 78089206029551605327761479038636, 230482287656166494197507811968614, 680274363439124716008660076469048, 2007847172373052260495864532377071, 5926212252399867616804722737789286, 17491366944520176989565419857250811, 51626216638453512501749734161720936, 152376098040498106446641567115125001, 449741948294412363423184360936143936, 1327424856370161746997340952807447813, 3917928394253003914310776410504946297, 11563865802896658210360917080640585035, 34131045504444072676025199024794308102, 100738653239527087596780295126775458778, 297332710045237647583837575056364384009, 877585093902734266671504982343304195401, 2590214836850932729875054483390580047805, 7645085300168406105043375077986516504441, 22564664681601737362604219809526854914655, 66600184589426223629692488678674281850269, 196572147201559446844327211538509564005457, 580187717100807114620112923748845144628646, 1712438877362864411873079434290663763303126, 5054307118663591535561067966277219646404563, 14917916655287589405860891358690352491721748, 44030612329103923073656416088674472069513113, 129957477768095494401253030877748897024366436, 383572817511819178378084729308651250250837950, 1132124975497756428229351306990005656258445796, 3341495803743449203159927529865766414538416128, 9862510277653711953998472241889224540072335266, 29109451182867070077872359881763068988818193678
Generating function in Maple syntax:
-(x^2+x-1)*(x-1)^4/(x^7-2*x^6+5*x^5+2*x^4-13*x^3+13*x^2-6*x+1)
Generating function in latex syntax:
-\frac{\left(x^{2}+x -1\right) \left(x -1\right)^{4}}{x^{7}-2 x^{6}+5 x^{5}+2 x^{4}-13 x^{3}+13 x^{2}-6 x +1}
Generating function in sympy syntax:
(x - 1)**4*(-x**2 - x + 1)/(x**7 - 2*x**6 + 5*x**5 + 2*x**4 - 13*x**3 + 13*x**2 - 6*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^7-2*x^6+5*x^5+2*x^4-13*x^3+13*x^2-6*x+1)*F(x)+(x^2+x-1)*(x-1)^4 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}-2 x^{6}+5 x^{5}+2 x^{4}-13 x^{3}+13 x^{2}-6 x +1\right) F \! \left(x \right)+\left(x^{2}+x -1\right) \left(x -1\right)^{4} = 0
Explicit closed form in Maple syntax:
-43659272/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+5)-43659272/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+5)-43659272/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+5)-43659272/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+5)-43659272/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+5)-43659272/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+5)-43659272/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 7)^(-n+5)+54338783/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+4)+54338783/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+4)+54338783/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+4)+54338783/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+4)+54338783/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+4)+54338783/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+4)+54338783/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 7)^(-n+4)-177868991/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+3)-177868991/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+3)-177868991/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+3)-177868991/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+3)-177868991/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+3)-177868991/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+3)-177868991/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 7)^(-n+3)-232906178/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+2)-232906178/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+2)-232906178/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+2)-232906178/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+2)-232906178/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+2)-232906178/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+2)-232906178/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 7)^(-n+2)+399007255/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+1)+399007255/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+1)+399007255/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+1)+399007255/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+1)+399007255/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+1)+399007255/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+1)+399007255/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 7)^(-n+1)+98285864/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n-1)+98285864/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n-1)+98285864/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n-1)+98285864/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n-1)+98285864/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n-1)+98285864/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n-1)+98285864/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 7)^(-n-1)-211124925/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n)-211124925/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n)-211124925/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n)-211124925/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n)-211124925/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n)-211124925/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n)-211124925/628571557*RootOf(_Z^7-2*_Z^6+5*_Z^5+2*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1,index = 7)^(-n)
Explicit closed form in latex syntax:
-\frac{43659272 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{628571557}-\frac{43659272 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{628571557}-\frac{43659272 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{628571557}-\frac{43659272 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{628571557}-\frac{43659272 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{628571557}-\frac{43659272 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{628571557}-\frac{43659272 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{628571557}+\frac{54338783 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{628571557}+\frac{54338783 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{628571557}+\frac{54338783 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{628571557}+\frac{54338783 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{628571557}+\frac{54338783 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{628571557}+\frac{54338783 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{628571557}+\frac{54338783 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{628571557}-\frac{177868991 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{628571557}-\frac{177868991 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{628571557}-\frac{177868991 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{628571557}-\frac{177868991 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{628571557}-\frac{177868991 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{628571557}-\frac{177868991 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{628571557}-\frac{177868991 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{628571557}-\frac{232906178 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{628571557}-\frac{232906178 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{628571557}-\frac{232906178 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{628571557}-\frac{232906178 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{628571557}-\frac{232906178 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{628571557}-\frac{232906178 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{628571557}-\frac{232906178 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{628571557}+\frac{399007255 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{628571557}+\frac{399007255 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{628571557}+\frac{399007255 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{628571557}+\frac{399007255 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{628571557}+\frac{399007255 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{628571557}+\frac{399007255 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{628571557}+\frac{399007255 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{628571557}+\frac{98285864 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{628571557}+\frac{98285864 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{628571557}+\frac{98285864 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{628571557}+\frac{98285864 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{628571557}+\frac{98285864 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{628571557}+\frac{98285864 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{628571557}+\frac{98285864 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{628571557}-\frac{211124925 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{628571557}-\frac{211124925 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{628571557}-\frac{211124925 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{628571557}-\frac{211124925 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{628571557}-\frac{211124925 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{628571557}-\frac{211124925 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{628571557}-\frac{211124925 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{628571557}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(6) = 194
a(n+7) = -a(n)+2*a(n+1)-5*a(n+2)-2*a(n+3)+13*a(n+4)-13*a(n+5)+6*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) = 20
a \! \left(5\right) = 64
a \! \left(6\right) = 194
a \! \left(n +7\right) = -a \! \left(n \right)+2 a \! \left(n +1\right)-5 a \! \left(n +2\right)-2 a \! \left(n +3\right)+13 a \! \left(n +4\right)-13 a \! \left(n +5\right)+6 a \! \left(n +6\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15789/
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[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[7,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[53,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[32,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[19,x]+F[24,x]+F[26,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[22,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[10,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[11,x]+F[29,x]
F[32,x] = F[33,x]+F[44,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[2,x]+F[33,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[2,x]+F[38,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[33,x]
F[44,x] = 2*F[19,x]+F[45,x]+F[47,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[32,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[37,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[38,x]+F[50,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[17,x]+F[55,x]
F[55,x] = F[38,x]+F[56,x]
F[56,x] = F[42,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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \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_{11}\! \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_{7}\! \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_{53}\! \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_{32}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{44}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{33}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{32}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{37}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{38}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{17}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{38}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{42}\! \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_6(x))
Eq(F_10(x), F_11(x) + F_14(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_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_7(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_53(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_32(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_19(x) + F_24(x) + F_26(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_22(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_10(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_11(x) + F_29(x))
Eq(F_32(x), F_33(x) + F_44(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_2(x) + F_33(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_2(x) + F_38(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_33(x))
Eq(F_44(x), 2*F_19(x) + F_45(x) + F_47(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_32(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_37(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_38(x) + F_50(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_17(x) + F_55(x))
Eq(F_55(x), F_38(x) + F_56(x))
Eq(F_56(x), F_42(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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, 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, 3, 0, 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": 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