0123_0213_0231_0312_2103
Counting sequence:
1, 1, 2, 6, 19, 56, 162, 471, 1376, 4023, 11758, 34360, 100412, 293450, 857604, 2506335, 7324721, 21406376, 62559799, 182830054, 534318038, 1561536297, 4563565967, 13336951844, 38977038132, 113909798865, 332899648108, 972894139222, 2843268268811, 8309408107872, 24284118336961, 70969965098462, 207408639514614, 606148582511478, 1771459978429976, 5177064742405728, 15129892672379533, 44216880349723633, 129223157771039054, 377652705759538487, 1103684267027441916, 3225500420641792008, 9426475736200467881, 27548731426764348533, 80510747013273155542, 235291429003436820214, 687635609111318463995, 2009604569620711991412, 5873038674451004806346, 17163865863475451859551, 50161135948405131398772, 146595154008322992144759, 428422099547911326480634, 1252057045286910244418512, 3659117599925012348129708, 10693715322701487814077258, 31252219771598517695162020, 91334135160571606874285495, 266922615625228663247552061, 780077269105962456845446212, 2279763909665138066912645591, 6662575221257591805279799218, 19471274367807602048373876310, 56904502075532093928451799353, 166302538565113842142271160467, 486016630046095878008113630304, 1420376181382688130871313510076, 4151027705467448112030594880017, 12131315096247616428737089735328, 35453583162214188243312709485038, 103612555528199201649444113538691, 302806111697221556947684592239732, 884946238549588137373252931503553, 2586241871848751985735653990060898, 7558252386796191609859133503309502, 22088877209954640345344498342506286, 64554406419242827814934726980202816, 188659267219917563826232279433371752, 551353828223671111791307979823080917, 1611323145565593503226794443732064869, 4709067293139599845464768681119971562, 13762177271731127480729450250761479167, 40219752971990104906961290125183063256, 117541614033026284234662396111077530688, 343513572525189317728683671311753818129, 1003913171345962456755756649822817743017, 2933920916699744983075896798876691712762, 8574339087420810941998793704982615288694, 25058375080120214371607820766093371641539, 73232718610020680504037734633702268394576, 214021502107260374313921037069261092073634, 625474572481327710654797541215521531078247, 1827939889070744654122180952566373064816168, 5342126418985187446010576510707534464135015, 15612282902216932219386409470037170843111686, 45626658431860484076025510561333086447167192, 133343212693260416344993196631821407852264444, 389693503369606005548928496650506107997621002, 1138873314218209617013643767284211460019786692, 3328339873832062346498295994935641466826226031, 9727022468117905914498978707383339151611942025
Generating function in Maple syntax:
(x-1)*(x^3+2*x-1)/(x^7+x^6+x^5+x^4-3*x^3+4*x^2-4*x+1)
Generating function in latex syntax:
\frac{\left(x -1\right) \left(x^{3}+2 x -1\right)}{x^{7}+x^{6}+x^{5}+x^{4}-3 x^{3}+4 x^{2}-4 x +1}
Generating function in sympy syntax:
(x - 1)*(x**3 + 2*x - 1)/(x**7 + x**6 + x**5 + x**4 - 3*x**3 + 4*x**2 - 4*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^7+x^6+x^5+x^4-3*x^3+4*x^2-4*x+1)*F(x)-(x-1)*(x^3+2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}+x^{6}+x^{5}+x^{4}-3 x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{3}+2 x -1\right) = 0
Explicit closed form in Maple syntax:
-4913210/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+5)-4913210/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+5)-4913210/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+5)-4913210/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+5)-4913210/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+5)-4913210/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+5)-4913210/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+5)-12379582/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+4)-12379582/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+4)-12379582/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+4)-12379582/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+4)-12379582/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+4)-12379582/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+4)-12379582/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+4)-17539116/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+3)-17539116/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+3)-17539116/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+3)-17539116/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+3)-17539116/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+3)-17539116/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+3)-17539116/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+3)-12857801/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+2)-12857801/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+2)-12857801/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+2)-12857801/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+2)-12857801/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+2)-12857801/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+2)-12857801/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+2)+8436448/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+1)+8436448/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+1)+8436448/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+1)+8436448/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+1)+8436448/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+1)+8436448/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+1)+8436448/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+1)+8632653/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n-1)+8632653/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n-1)+8632653/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n-1)+8632653/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n-1)+8632653/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n-1)+8632653/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n-1)+8632653/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n-1)+16507266/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n)+16507266/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n)+16507266/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n)+16507266/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n)+16507266/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n)+16507266/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n)+16507266/163223773*RootOf(_Z^7+_Z^6+_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n)
Explicit closed form in latex syntax:
-\frac{4913210 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{163223773}-\frac{4913210 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{163223773}-\frac{4913210 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{163223773}-\frac{4913210 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{163223773}-\frac{4913210 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{163223773}-\frac{4913210 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{163223773}-\frac{4913210 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{163223773}-\frac{12379582 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{163223773}-\frac{12379582 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{163223773}-\frac{12379582 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{163223773}-\frac{12379582 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{163223773}-\frac{12379582 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{163223773}-\frac{12379582 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{163223773}-\frac{12379582 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{163223773}-\frac{17539116 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{163223773}-\frac{17539116 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{163223773}-\frac{17539116 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{163223773}-\frac{17539116 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{163223773}-\frac{17539116 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{163223773}-\frac{17539116 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{163223773}-\frac{17539116 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{163223773}-\frac{12857801 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{163223773}-\frac{12857801 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{163223773}-\frac{12857801 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{163223773}-\frac{12857801 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{163223773}-\frac{12857801 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{163223773}-\frac{12857801 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{163223773}-\frac{12857801 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{163223773}+\frac{8436448 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{163223773}+\frac{8436448 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{163223773}+\frac{8436448 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{163223773}+\frac{8436448 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{163223773}+\frac{8436448 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{163223773}+\frac{8436448 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{163223773}+\frac{8436448 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{163223773}+\frac{8632653 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{163223773}+\frac{8632653 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{163223773}+\frac{8632653 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{163223773}+\frac{8632653 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{163223773}+\frac{8632653 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{163223773}+\frac{8632653 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{163223773}+\frac{8632653 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{163223773}+\frac{16507266 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{163223773}+\frac{16507266 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{163223773}+\frac{16507266 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{163223773}+\frac{16507266 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{163223773}+\frac{16507266 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{163223773}+\frac{16507266 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{163223773}+\frac{16507266 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{163223773}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 162
a(n+7) = -a(n)-a(n+1)-a(n+2)-a(n+3)+3*a(n+4)-4*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) = 56
a \! \left(6\right) = 162
a \! \left(n +7\right) = -a \! \left(n \right)-a \! \left(n +1\right)-a \! \left(n +2\right)-a \! \left(n +3\right)+3 a \! \left(n +4\right)-4 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/16681/
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[14,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[11,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]+F[17,x]+F[58,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[30,x]
F[19,x] = F[20,x]+F[7,x]
F[20,x] = F[16,x]+F[21,x]+F[27,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]
F[23,x] = F[11,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[11,x]+F[24,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[13,x]+F[29,x]
F[29,x] = F[24,x]
F[30,x] = F[31,x]+F[38,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[2,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[34,x]*F[4,x]
F[37,x] = F[35,x]
F[38,x] = 2*F[16,x]+F[39,x]+F[54,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[35,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[42,x]+F[45,x]
F[45,x] = F[16,x]+F[46,x]+F[52,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[50,x]
F[48,x] = F[11,x]+F[49,x]
F[49,x] = F[16,x]+F[21,x]+F[25,x]
F[50,x] = F[35,x]+F[51,x]
F[51,x] = 2*F[16,x]+F[39,x]+F[43,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[2,x]+F[45,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[45,x]
F[57,x] = F[42,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[53,x]+F[60,x]
F[60,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_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{58}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{30}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{27}\! \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_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \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_{11}\! \left(x \right)+F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{24}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{38}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{39}\! \left(x \right)+F_{54}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{35}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{16}\! \left(x \right)+F_{46}\! \left(x \right)+F_{52}\! \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_{11}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{50}\! \left(x \right) = F_{35}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{39}\! \left(x \right)+F_{43}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{2}\! \left(x \right)+F_{45}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{45}\! \left(x \right)
F_{57}\! \left(x \right) = F_{42}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{53}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \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_14(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_11(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_58(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_30(x))
Eq(F_19(x), F_20(x) + F_7(x))
Eq(F_20(x), F_16(x) + F_21(x) + F_27(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_11(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_11(x) + F_24(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_13(x) + F_29(x))
Eq(F_29(x), F_24(x))
Eq(F_30(x), F_31(x) + F_38(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_2(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_34(x)*F_4(x))
Eq(F_37(x), F_35(x))
Eq(F_38(x), 2*F_16(x) + F_39(x) + F_54(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_35(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_42(x) + F_45(x))
Eq(F_45(x), F_16(x) + F_46(x) + F_52(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_11(x) + F_49(x))
Eq(F_49(x), F_16(x) + F_21(x) + F_25(x))
Eq(F_50(x), F_35(x) + F_51(x))
Eq(F_51(x), 2*F_16(x) + F_39(x) + F_43(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_2(x) + F_45(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_45(x))
Eq(F_57(x), F_42(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_53(x) + F_60(x))
Eq(F_60(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:
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