0132_0321_1203_2013
Counting sequence:
1, 1, 2, 6, 20, 63, 192, 581, 1762, 5356, 16294, 49572, 150802, 458733, 1395443, 4244899, 12912922, 39280973, 119492313, 363494342, 1105745889, 3363667204, 10232239976, 31126365825, 94686076003, 288034043015, 876196516393, 2665380548074, 8108059475130, 24664631286430, 75029547868656, 228239092166035, 694300907745414, 2112056028272810, 6424852130832585, 19544332323815831, 59453652505203066, 180857382981688871, 550166248846089757, 1673599918229646711, 5091073274983594502, 15486991131470492597, 47111263451029196014, 143311965837068445016, 435953486440330207948, 1326165901286662120773, 4034182664980668372816, 12271941058536255875032, 37331115085962433492080, 113560857806759710539297, 345450930038154374492583, 1050858080583487209736190, 3196699182154880408801371, 9724325149134944969767124, 29581294397039329167950869, 89985985123312463296667257, 273736416328808633337626242, 832703287315865423204487008, 2533074605147719630945064241, 7705586194966404008135359281, 23440311820027989342049840497, 71305180984032771369341295502, 216909607440520711806170103590, 659836734872555223513925078947, 2007216378401600012280885931583, 6105931023227744576492009593056, 18574177682878403602457141476370, 56502452334085270786768148970309, 171879863231223189589468979027672, 522856728584190604681245614267001, 1590524646031372382169043889955267, 4838359174383041387085859964080731, 14718237506565993648043785898675505, 44772723043511724203324763666451134, 136198150616657025685629816211863165, 414313335674716611504701838935914840, 1260336791217170478497074463009724357, 3833931207424874451528807844044141460, 11662778239672561106575020972871432142, 35478048224850709199730335053357024461, 107923847987027405192352192057869458478, 328303205703644988156800987135983417897, 998694884269188577795528443576227975782, 3038019289905381580902074448431317928297, 9241622592861364545546198550798054341912, 28112918319075456048357448750739116910065, 85519200602878845395564894922484208009865, 260148504995049983860576821240040023937553, 791369004551725398007104039656721266557022, 2407336153544703731902034691456071500405760, 7323091153217663805681425470350400877491464, 22276765942874358371621847181986359813169275, 67765686714900401525546920259705313693737173, 206142503257342270356772693990534851332427293, 627083317667606182777769640973554572019121687, 1907580829200035021706756359344226582992049635, 5802840734889907514196602188890705697636745696, 17652180226941559207538858172529134797848223573, 53697745811101995554789436968065296119733291923, 163347975611130049199928686579479954673653395621, 496902816556178872406750645086278795478058965087
Generating function in Maple syntax:
-(x^3-2*x^2+3*x-1)*(x-1)^2/(x^7-x^6-x^5+9*x^4-15*x^3+13*x^2-6*x+1)
Generating function in latex syntax:
-\frac{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{2}}{x^{7}-x^{6}-x^{5}+9 x^{4}-15 x^{3}+13 x^{2}-6 x +1}
Generating function in sympy syntax:
(x - 1)**2*(-x**3 + 2*x**2 - 3*x + 1)/(x**7 - x**6 - x**5 + 9*x**4 - 15*x**3 + 13*x**2 - 6*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^7-x^6-x^5+9*x^4-15*x^3+13*x^2-6*x+1)*F(x)+(x^3-2*x^2+3*x-1)*(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}-x^{6}-x^{5}+9 x^{4}-15 x^{3}+13 x^{2}-6 x +1\right) F \! \left(x \right)+\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
-2001919/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+5)-2001919/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+5)-2001919/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+5)-2001919/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+5)-2001919/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+5)-2001919/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+5)-2001919/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 7)^(-n+5)+229205/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+4)+229205/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+4)+229205/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+4)+229205/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+4)+229205/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+4)+229205/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+4)+229205/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 7)^(-n+4)+2957334/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+3)+2957334/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+3)+2957334/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+3)+2957334/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+3)+2957334/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+3)+2957334/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+3)+2957334/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 7)^(-n+3)-14948575/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+2)-14948575/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+2)-14948575/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+2)-14948575/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+2)-14948575/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+2)-14948575/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+2)-14948575/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 7)^(-n+2)+15057532/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 7)^(-n+1)+15057532/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+1)+15057532/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+1)+15057532/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+1)+15057532/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+1)+15057532/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+1)+15057532/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+1)+2276247/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n-1)+2276247/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n-1)+2276247/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n-1)+2276247/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n-1)+2276247/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n-1)+2276247/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n-1)+2276247/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 7)^(-n-1)-6330956/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n)-6330956/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n)-6330956/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n)-6330956/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n)-6330956/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n)-6330956/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n)-6330956/16771681*RootOf(_Z^7-_Z^6-_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 7)^(-n)
Explicit closed form in latex syntax:
-\frac{2001919 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{16771681}-\frac{2001919 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{16771681}-\frac{2001919 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{16771681}-\frac{2001919 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{16771681}-\frac{2001919 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{16771681}-\frac{2001919 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{16771681}-\frac{2001919 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{16771681}+\frac{229205 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{16771681}+\frac{229205 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{16771681}+\frac{229205 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{16771681}+\frac{229205 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{16771681}+\frac{229205 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{16771681}+\frac{229205 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{16771681}+\frac{229205 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{16771681}+\frac{2957334 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{16771681}+\frac{2957334 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{16771681}+\frac{2957334 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{16771681}+\frac{2957334 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{16771681}+\frac{2957334 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{16771681}+\frac{2957334 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{16771681}+\frac{2957334 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{16771681}-\frac{14948575 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{16771681}-\frac{14948575 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{16771681}-\frac{14948575 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{16771681}-\frac{14948575 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{16771681}-\frac{14948575 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{16771681}-\frac{14948575 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{16771681}-\frac{14948575 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{16771681}+\frac{15057532 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{16771681}+\frac{15057532 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{16771681}+\frac{15057532 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{16771681}+\frac{15057532 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{16771681}+\frac{15057532 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{16771681}+\frac{15057532 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{16771681}+\frac{15057532 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{16771681}+\frac{2276247 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{16771681}+\frac{2276247 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{16771681}+\frac{2276247 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{16771681}+\frac{2276247 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{16771681}+\frac{2276247 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{16771681}+\frac{2276247 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{16771681}+\frac{2276247 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{16771681}-\frac{6330956 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{16771681}-\frac{6330956 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{16771681}-\frac{6330956 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{16771681}-\frac{6330956 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{16771681}-\frac{6330956 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{16771681}-\frac{6330956 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{16771681}-\frac{6330956 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-\textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{16771681}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(6) = 192
a(n+7) = -a(n)+a(n+1)+a(n+2)-9*a(n+3)+15*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) = 63
a \! \left(6\right) = 192
a \! \left(n +7\right) = -a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)-9 a \! \left(n +3\right)+15 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/15430/
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[21,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[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[11,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[11,x]+F[20,x]
F[20,x] = F[18,x]
F[21,x] = F[2,x]+F[22,x]
F[22,x] = F[15,x]+F[23,x]+F[54,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[29,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[19,x]
F[29,x] = F[22,x]+F[30,x]
F[30,x] = 2*F[15,x]+F[31,x]+F[35,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[26,x]
F[34,x] = F[30,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]+F[53,x]
F[38,x] = F[15,x]+F[39,x]+F[50,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[43,x]
F[41,x] = F[11,x]+F[42,x]
F[42,x] = F[18,x]
F[43,x] = F[38,x]+F[44,x]
F[44,x] = 2*F[15,x]+F[45,x]+F[49,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[42,x]
F[48,x] = F[44,x]
F[49,x] = F[37,x]*F[4,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[2,x]+F[52,x]
F[52,x] = F[50,x]
F[53,x] = F[49,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[51,x]+F[56,x]
F[56,x] = F[38,x]+F[57,x]
F[57,x] = 2*F[15,x]+F[49,x]+F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[52,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_{21}\! \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_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{54}\! \left(x \right)
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_{29}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{19}\! \left(x \right)
F_{29}\! \left(x \right) = F_{22}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{26}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{53}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{39}\! \left(x \right)+F_{50}\! \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_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{11}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)
F_{43}\! \left(x \right) = F_{38}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{42}\! \left(x \right)
F_{48}\! \left(x \right) = F_{44}\! \left(x \right)
F_{49}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{2}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{38}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{52}\! \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_21(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) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_11(x) + F_20(x))
Eq(F_20(x), F_18(x))
Eq(F_21(x), F_2(x) + F_22(x))
Eq(F_22(x), F_15(x) + F_23(x) + F_54(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_29(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_19(x))
Eq(F_29(x), F_22(x) + F_30(x))
Eq(F_30(x), 2*F_15(x) + F_31(x) + F_35(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_26(x))
Eq(F_34(x), F_30(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x) + F_53(x))
Eq(F_38(x), F_15(x) + F_39(x) + F_50(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_43(x))
Eq(F_41(x), F_11(x) + F_42(x))
Eq(F_42(x), F_18(x))
Eq(F_43(x), F_38(x) + F_44(x))
Eq(F_44(x), 2*F_15(x) + F_45(x) + F_49(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_42(x))
Eq(F_48(x), F_44(x))
Eq(F_49(x), F_37(x)*F_4(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_2(x) + F_52(x))
Eq(F_52(x), F_50(x))
Eq(F_53(x), F_49(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_51(x) + F_56(x))
Eq(F_56(x), F_38(x) + F_57(x))
Eq(F_57(x), 2*F_15(x) + F_49(x) + F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_52(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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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