0123_0213_0321_1302_2103
Counting sequence:
1, 1, 2, 6, 19, 51, 123, 311, 826, 2186, 5680, 14684, 38174, 99598, 259589, 675469, 1757398, 4574974, 11912569, 31013917, 80732941, 210161005, 547113192, 1424317400, 3707907096, 9652657392, 25128498788, 65416634948, 170298124829, 443333735421, 1154121311174, 3004500994962, 7821560279315, 20361717532539, 53007259619935, 137992755956979, 359233845262234, 935186471967018, 2434552695400304, 6337823449256164, 16499131929011226, 42951867934670042, 111815759111049025, 291087781920637065, 757783136998629138, 1972722039718101778, 5135548757535409561, 13369273780418785093, 34803969317857191089, 90604493565166077225, 235868908502091162448, 614032922769629350000, 1598499915217567662656, 4161343609049552316080, 10833144542380149377736, 28201713605538553162824, 73416970223123257204409, 191124964678698985593401, 497551887695366328089722, 1295267111574869737006990, 3371943573763492315426307, 8778114848312682029420083, 22851894939675503361299635, 59489891777196032326489439, 154868873369362220425765114, 403167113305967581343768458, 1049557071831993425937431104, 2732291426251674234027372924, 7112920905708861760878621110, 18516928071716651298224342086, 48204757195870825720116597533, 125490502923222512574277103909, 326686975311064198647329957854, 850457822319732452962424085942, 2213980238594244317232387961513, 5763611513991124796220866938397, 15004297285554562643749532592293, 39060394075277478812874971921909, 101685161009496831721800884861688, 264714993648047120011864344165256, 689127372828185829550525948608312, 1793991830370130144846565570371712, 4670263893634634797264058888516652, 12158006779600158285852602012421260, 31650701591888990895068418496856341, 82395653285838151357896884415119477, 214498994933491605153659912886256798, 558401044140832709894228060494725498, 1453674532107965169438052176540234851, 3784322517789481539195488345952428123, 9851652899140758590391598151538743223, 25646615580175452629299333075107454139, 66765333437058649098946066677380983578, 173808888546187132420653983429556245866, 452473284899268575496937287992275044576, 1177914865344358715277329784471506024676, 3066442763152538104051326228479510755826, 7982810554769272321367539474170154606290, 20781494805342866760596122960973833082329, 54100059544376214598343050802946143393057, 140837628386220574899679574427300080194874
Generating function in Maple syntax:
-1/(x^7+7*x^6+12*x^5+8*x^4+3*x^3+x^2+x-1)
Generating function in latex syntax:
-\frac{1}{x^{7}+7 x^{6}+12 x^{5}+8 x^{4}+3 x^{3}+x^{2}+x -1}
Generating function in sympy syntax:
-1/(x**7 + 7*x**6 + 12*x**5 + 8*x**4 + 3*x**3 + x**2 + x - 1)
Implicit equation for the generating function in Maple syntax:
(x^7+7*x^6+12*x^5+8*x^4+3*x^3+x^2+x-1)*F(x)+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}+7 x^{6}+12 x^{5}+8 x^{4}+3 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+1 = 0
Explicit closed form in Maple syntax:
-318311/197428886*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 1)^(-n+5)-318311/197428886*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 2)^(-n+5)-318311/197428886*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 3)^(-n+5)-318311/197428886*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 4)^(-n+5)-318311/197428886*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 5)^(-n+5)-318311/197428886*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 6)^(-n+5)-318311/197428886*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 7)^(-n+5)+21563/10390994*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 1)^(-n+4)+21563/10390994*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 2)^(-n+4)+21563/10390994*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 3)^(-n+4)+21563/10390994*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 4)^(-n+4)+21563/10390994*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 5)^(-n+4)+21563/10390994*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 6)^(-n+4)+21563/10390994*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 7)^(-n+4)+13854277/197428886*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 1)^(-n+3)+13854277/197428886*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 2)^(-n+3)+13854277/197428886*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 3)^(-n+3)+13854277/197428886*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 4)^(-n+3)+13854277/197428886*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 5)^(-n+3)+13854277/197428886*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 6)^(-n+3)+13854277/197428886*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 7)^(-n+3)+1331685/10390994*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 1)^(-n+2)+1331685/10390994*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 2)^(-n+2)+1331685/10390994*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 3)^(-n+2)+1331685/10390994*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 4)^(-n+2)+1331685/10390994*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 5)^(-n+2)+1331685/10390994*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 6)^(-n+2)+1331685/10390994*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 7)^(-n+2)+13048662/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 7)^(-n+1)+13048662/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 1)^(-n+1)+13048662/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 2)^(-n+1)+13048662/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 3)^(-n+1)+13048662/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 4)^(-n+1)+13048662/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 5)^(-n+1)+13048662/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 6)^(-n+1)+4957502/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 1)^(-n-1)+4957502/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 2)^(-n-1)+4957502/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 3)^(-n-1)+4957502/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 4)^(-n-1)+4957502/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 5)^(-n-1)+4957502/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 6)^(-n-1)+4957502/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 7)^(-n-1)+18813840/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 1)^(-n)+18813840/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 2)^(-n)+18813840/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 3)^(-n)+18813840/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 4)^(-n)+18813840/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 5)^(-n)+18813840/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 6)^(-n)+18813840/98714443*RootOf(_Z^7+7*_Z^6+12*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 7)^(-n)
Explicit closed form in latex syntax:
-\frac{318311 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{197428886}-\frac{318311 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{197428886}-\frac{318311 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{197428886}-\frac{318311 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{197428886}-\frac{318311 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{197428886}-\frac{318311 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{197428886}-\frac{318311 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{197428886}+\frac{21563 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{10390994}+\frac{21563 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{10390994}+\frac{21563 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{10390994}+\frac{21563 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{10390994}+\frac{21563 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{10390994}+\frac{21563 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{10390994}+\frac{21563 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{10390994}+\frac{13854277 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{197428886}+\frac{13854277 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{197428886}+\frac{13854277 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{197428886}+\frac{13854277 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{197428886}+\frac{13854277 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{197428886}+\frac{13854277 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{197428886}+\frac{13854277 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{197428886}+\frac{1331685 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{10390994}+\frac{1331685 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{10390994}+\frac{1331685 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{10390994}+\frac{1331685 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{10390994}+\frac{1331685 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{10390994}+\frac{1331685 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{10390994}+\frac{1331685 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{10390994}+\frac{13048662 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{98714443}+\frac{13048662 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{98714443}+\frac{13048662 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{98714443}+\frac{13048662 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{98714443}+\frac{13048662 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{98714443}+\frac{13048662 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{98714443}+\frac{13048662 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{98714443}+\frac{4957502 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{98714443}+\frac{4957502 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{98714443}+\frac{4957502 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{98714443}+\frac{4957502 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{98714443}+\frac{4957502 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{98714443}+\frac{4957502 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{98714443}+\frac{4957502 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{98714443}+\frac{18813840 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{98714443}+\frac{18813840 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{98714443}+\frac{18813840 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{98714443}+\frac{18813840 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{98714443}+\frac{18813840 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{98714443}+\frac{18813840 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{98714443}+\frac{18813840 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+7 \textit{\_Z}^{6}+12 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{98714443}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 123
a(n+7) = a(n)+7*a(n+1)+12*a(n+2)+8*a(n+3)+3*a(n+4)+a(n+5)+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) = 51
a \! \left(6\right) = 123
a \! \left(n +7\right) = a \! \left(n \right)+7 a \! \left(n +1\right)+12 a \! \left(n +2\right)+8 a \! \left(n +3\right)+3 a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16843/
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[20,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[15,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[62,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[36,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[22,x]+F[27,x]+F[34,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[33,x]
F[29,x] = F[11,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[4,x]
F[33,x] = F[17,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[14,x]
F[36,x] = F[37,x]+F[52,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[46,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[2,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[2,x]*F[4,x]
F[46,x] = F[47,x]+F[49,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[43,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[47,x]
F[52,x] = 2*F[22,x]+F[53,x]+F[60,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[59,x]
F[55,x] = F[41,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[44,x]
F[59,x] = F[49,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[46,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[72,x]
F[64,x] = F[2,x]+F[65,x]
F[65,x] = F[22,x]+F[42,x]+F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]+F[70,x]
F[68,x] = F[11,x]+F[69,x]
F[69,x] = F[18,x]+F[22,x]+F[27,x]
F[70,x] = F[41,x]+F[71,x]
F[71,x] = 2*F[22,x]+F[50,x]+F[53,x]
F[72,x] = F[47,x]+F[73,x]
F[73,x] = F[50,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_{20}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{62}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{36}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{33}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{17}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{52}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{46}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \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_{2}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{49}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{47}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{53}\! \left(x \right)+F_{60}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{59}\! \left(x \right)
F_{55}\! \left(x \right) = F_{41}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{44}\! \left(x \right)
F_{59}\! \left(x \right) = F_{49}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{46}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{72}\! \left(x \right)
F_{64}\! \left(x \right) = F_{2}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{22}\! \left(x \right)+F_{42}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{68}\! \left(x \right) = F_{11}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)+F_{27}\! \left(x \right)
F_{70}\! \left(x \right) = F_{41}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{72}\! \left(x \right) = F_{47}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{50}\! \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_20(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_15(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_62(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_36(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_34(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_33(x))
Eq(F_29(x), F_11(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_4(x))
Eq(F_33(x), F_17(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_14(x))
Eq(F_36(x), F_37(x) + F_52(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_46(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_2(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_2(x)*F_4(x))
Eq(F_46(x), F_47(x) + F_49(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_43(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_47(x))
Eq(F_52(x), 2*F_22(x) + F_53(x) + F_60(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_59(x))
Eq(F_55(x), F_41(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_44(x))
Eq(F_59(x), F_49(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_46(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_72(x))
Eq(F_64(x), F_2(x) + F_65(x))
Eq(F_65(x), F_22(x) + F_42(x) + F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_70(x))
Eq(F_68(x), F_11(x) + F_69(x))
Eq(F_69(x), F_18(x) + F_22(x) + F_27(x))
Eq(F_70(x), F_41(x) + F_71(x))
Eq(F_71(x), 2*F_22(x) + F_50(x) + F_53(x))
Eq(F_72(x), F_47(x) + F_73(x))
Eq(F_73(x), F_50(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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