0123_0132_1203_2031_2103
Counting sequence:
1, 1, 2, 6, 19, 55, 158, 455, 1310, 3768, 10834, 31147, 89540, 257396, 739909, 2126919, 6113947, 17574841, 50519686, 145221070, 417444264, 1199961469, 3449340540, 9915276479, 28501884556, 81929880376, 235510927512, 676986181296, 1946025579269, 5593933314445, 16079999281968, 46222641982980, 132868950692410, 381937450999698, 1097895450466434, 3155947176661247, 9071904412754580, 26077575151631596, 74961099108594724, 215478868218837004, 619403173125746792, 1780500770444679096, 5118125206811414241, 14712268631063361495, 42291042037132095849, 121567399388709167039, 349450660997144983855, 1004510790601670295109, 2887508999284552463061, 8300267452533036726176, 23859471884122074323547, 68585066908354041982786, 197150692423906681297269, 566718052125884379939260, 1629055148915172142497103, 4682788325256213568963230, 13460874245883342074956310, 38693855642849613944106753, 111227134074710561382870209, 319727128479113237284514172, 919069231945104371900769970, 2641903604258419604383760221, 7594264296523115564778450889, 21830035778930116527037414089, 62751366492149643626031826812, 180381472413008945889574521501, 518514215845094445349313088171, 1490491170943914928283639244145, 4284480276092280613810838588634, 12315920814612141313782449790209, 35402638298556987755090473083570, 101766389810774776982600333716264, 292531816634148742027023844785391, 840895151163304457923968366022026, 2417188883540446639478253935897107, 6948312272497359825718004455922601, 19973219207190506322310801614807144, 57413868268056506025310129282290500, 165038606711682943620089295219746188, 474410495704012460272000408298769705, 1363713151234417450783063136517033385, 3920051465324223161363033284909162558, 11268354695326312212327961398037877409, 32391364925404242310826773726897212297, 93110356400644444717104705371338247835, 267649680370696345062857833159345066021, 769370391992615117079353383430808281654, 2211587920654501378418241550336219307049, 6357303558455429913612842736731202083507, 18274339517277473364444621624280407320806, 52530366329379104037304406255191341559101, 151000772645701041986075208526349941120020, 434058144514527336794941892404894018949619, 1247718601158814585954794912008555591144479, 3586620196745565241855242753242924652199560, 10309892329693525056876466944180399374515319, 29636224082584082731628313196544047112967671, 85190586844784783161376011157497350894915676, 244883965876870834112931999148652954219039692, 703929377230902165232628046334690209402450741, 2023474939873501335051209298304127330336842644
Generating function in Maple syntax:
(x^3+x^2+x-1)^2/(x^7+2*x^6+2*x^5+2*x^4-3*x+1)
Generating function in latex syntax:
\frac{\left(x^{3}+x^{2}+x -1\right)^{2}}{x^{7}+2 x^{6}+2 x^{5}+2 x^{4}-3 x +1}
Generating function in sympy syntax:
(x**3 + x**2 + x - 1)**2/(x**7 + 2*x**6 + 2*x**5 + 2*x**4 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^7+2*x^6+2*x^5+2*x^4-3*x+1)*F(x)-(x^3+x^2+x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}+2 x^{6}+2 x^{5}+2 x^{4}-3 x +1\right) F \! \left(x \right)-\left(x^{3}+x^{2}+x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
-3154876/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 1)^(-n+5)-3154876/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 2)^(-n+5)-3154876/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 3)^(-n+5)-3154876/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 4)^(-n+5)-3154876/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 5)^(-n+5)-3154876/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 6)^(-n+5)-3154876/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 7)^(-n+5)-9803891/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 1)^(-n+4)-9803891/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 2)^(-n+4)-9803891/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 3)^(-n+4)-9803891/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 4)^(-n+4)-9803891/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 5)^(-n+4)-9803891/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 6)^(-n+4)-9803891/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 7)^(-n+4)-17787108/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 1)^(-n+3)-17787108/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 2)^(-n+3)-17787108/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 3)^(-n+3)-17787108/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 4)^(-n+3)-17787108/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 5)^(-n+3)-17787108/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 6)^(-n+3)-17787108/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 7)^(-n+3)-15245535/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 1)^(-n+2)-15245535/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 2)^(-n+2)-15245535/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 3)^(-n+2)-15245535/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 4)^(-n+2)-15245535/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 5)^(-n+2)-15245535/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 6)^(-n+2)-15245535/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 7)^(-n+2)-4267606/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 1)^(-n+1)-4267606/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 2)^(-n+1)-4267606/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 3)^(-n+1)-4267606/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 4)^(-n+1)-4267606/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 5)^(-n+1)-4267606/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 6)^(-n+1)-4267606/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 7)^(-n+1)+5483219/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 1)^(-n-1)+5483219/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 2)^(-n-1)+5483219/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 3)^(-n-1)+5483219/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 4)^(-n-1)+5483219/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 5)^(-n-1)+5483219/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 6)^(-n-1)+5483219/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 7)^(-n-1)+6087276/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 1)^(-n)+6087276/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 2)^(-n)+6087276/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 3)^(-n)+6087276/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 4)^(-n)+6087276/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 5)^(-n)+6087276/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 6)^(-n)+6087276/62597401*RootOf(_Z^7+2*_Z^6+2*_Z^5+2*_Z^4-3*_Z+1,index = 7)^(-n)
Explicit closed form in latex syntax:
-\frac{3154876 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{62597401}-\frac{3154876 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{62597401}-\frac{3154876 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{62597401}-\frac{3154876 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{62597401}-\frac{3154876 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{62597401}-\frac{3154876 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{62597401}-\frac{3154876 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{62597401}-\frac{9803891 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{62597401}-\frac{9803891 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{62597401}-\frac{9803891 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{62597401}-\frac{9803891 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{62597401}-\frac{9803891 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{62597401}-\frac{9803891 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{62597401}-\frac{9803891 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{62597401}-\frac{17787108 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{62597401}-\frac{17787108 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{62597401}-\frac{17787108 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{62597401}-\frac{17787108 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{62597401}-\frac{17787108 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{62597401}-\frac{17787108 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{62597401}-\frac{17787108 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{62597401}-\frac{15245535 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{62597401}-\frac{15245535 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{62597401}-\frac{15245535 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{62597401}-\frac{15245535 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{62597401}-\frac{15245535 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{62597401}-\frac{15245535 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{62597401}-\frac{15245535 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{62597401}-\frac{4267606 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{62597401}-\frac{4267606 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{62597401}-\frac{4267606 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{62597401}-\frac{4267606 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{62597401}-\frac{4267606 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{62597401}-\frac{4267606 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{62597401}-\frac{4267606 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{62597401}+\frac{5483219 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{62597401}+\frac{5483219 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{62597401}+\frac{5483219 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{62597401}+\frac{5483219 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{62597401}+\frac{5483219 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{62597401}+\frac{5483219 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{62597401}+\frac{5483219 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{62597401}+\frac{6087276 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{62597401}+\frac{6087276 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{62597401}+\frac{6087276 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{62597401}+\frac{6087276 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{62597401}+\frac{6087276 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{62597401}+\frac{6087276 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{62597401}+\frac{6087276 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}+2 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{62597401}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 158
a(n+3) = -1/2*a(n)-a(n+1)-a(n+2)+3/2*a(n+6)-1/2*a(n+7), 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) = 55
a \! \left(6\right) = 158
a \! \left(n +3\right) = -\frac{a \! \left(n \right)}{2}-a \! \left(n +1\right)-a \! \left(n +2\right)+\frac{3 a \! \left(n +6\right)}{2}-\frac{a \! \left(n +7\right)}{2}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16440/
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[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[44,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[27,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[16,x]
F[27,x] = F[28,x]+F[41,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[2,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[2,x]*F[4,x]
F[34,x] = F[28,x]+F[35,x]
F[35,x] = 2*F[13,x]+F[36,x]+F[40,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[32,x]
F[39,x] = F[40,x]
F[40,x] = F[28,x]*F[4,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[38,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[31,x]+F[46,x]
F[46,x] = F[20,x]+F[47,x]
F[47,x] = 2*F[13,x]+F[48,x]+F[51,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[38,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[20,x]*F[4,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{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_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{44}\! \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_{27}\! \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_{16}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{41}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{28}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{32}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \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_{38}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{31}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{20}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{38}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_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_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_44(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_27(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_16(x))
Eq(F_27(x), F_28(x) + F_41(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_34(x))
Eq(F_31(x), F_2(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_2(x)*F_4(x))
Eq(F_34(x), F_28(x) + F_35(x))
Eq(F_35(x), 2*F_13(x) + F_36(x) + F_40(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_32(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_28(x)*F_4(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_38(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_31(x) + F_46(x))
Eq(F_46(x), F_20(x) + F_47(x))
Eq(F_47(x), 2*F_13(x) + F_48(x) + F_51(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_38(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_20(x)*F_4(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "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]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "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]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "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]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [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": [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]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [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": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 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": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 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