0132_0231_1302_2013_2103
Counting sequence:
1, 1, 2, 6, 19, 55, 151, 412, 1133, 3133, 8672, 23986, 66304, 183258, 506540, 1400205, 3870589, 10699421, 29576085, 81756100, 225995506, 624711825, 1726870460, 4773531428, 13195315639, 36475375166, 100827674741, 278714612624, 770441604919, 2129706301146, 5887076836654, 16273452188825, 44984166756699, 124348247405174, 343731756041779, 950166347961786, 2626513474316790, 7260384505867174, 20069641251962494, 55477846890499751, 153355581047377194, 423915771006768950, 1171816373952113114, 3239213325328933023, 8954050481135486690, 24751386205964465545, 68419439940338773687, 189129599562451024000, 522804709624110345385, 1445171803025457934947, 3994840715496751815573, 11042806335399745479787, 30525265072047129392001, 84379982716142594139995, 233248801160992914358696, 644761962396471060127985, 1782294211520562632605388, 4926737062175506233171979, 13618816648181488670011399, 37646045355399192234127239, 104063720623628055745885260, 287659908173578715640481116, 795168789608349312499146504, 2198058839626935649644454103, 6076021500846114460623065595, 16795745688504937048577070396, 46427925443260666994504043764, 128339181894153004283013424819, 354763764523348123530105308751, 980661764873783427044446502865, 2710810948737892710303461664848, 7493405232071045318855671870485, 20713772754301704492661847548307, 57258398342120206311621828046824, 158277500655888529773187237017368, 437521270926754298947672975207816, 1209425608315230860090949321982892, 3343175290541564163740106649673632, 9241429110185079125388414798499185, 25545777464974475126657258056316271, 70615349477796112601049955981493403, 195199679818249843965980486725079997, 539584032125028546789637286991825058, 1491554330393339123719594745646173022, 4123054404989550813558904557157053821, 11397223205420056745887562549888229588, 31504967927895832432550400941400482664, 87088142984312384491320053946416664425, 240734879204258403443603346322099296550, 665455480844600106340691919158795849377, 1839496621552647117666301156497400228144, 5084859796193923488041037373987592570137, 14055910102800537249384902298970540063474, 38854288365215615966772869272033994647828, 107403626896172431074488736299677465447059, 296892300845211975353114848754731740669679, 820689587944493347262832784826804949947584, 2268605140123371499668151011160802353106149, 6271030310844233518333429289201776683382588, 17334802105486062476646792991364017048523096, 47918021304526277276822235218189233134199452
Generating function in Maple syntax:
(x-1)^3/(x^7+x^4+4*x^3-5*x^2+4*x-1)
Generating function in latex syntax:
\frac{\left(x -1\right)^{3}}{x^{7}+x^{4}+4 x^{3}-5 x^{2}+4 x -1}
Generating function in sympy syntax:
(x - 1)**3/(x**7 + x**4 + 4*x**3 - 5*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^7+x^4+4*x^3-5*x^2+4*x-1)*F(x)-(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}+x^{4}+4 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
-2573401/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+5)-2573401/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+5)-2573401/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+5)-2573401/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+5)-2573401/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+5)-2573401/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+5)-2573401/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+5)-50727/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+4)-50727/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+4)-50727/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+4)-50727/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+4)-50727/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+4)-50727/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+4)-50727/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+4)+3588996/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+3)+3588996/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+3)+3588996/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+3)+3588996/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+3)+3588996/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+3)+3588996/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+3)+3588996/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+3)+1780405/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+2)+1780405/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+2)+1780405/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+2)+1780405/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+2)+1780405/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+2)+1780405/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+2)+1780405/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+2)-9255116/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+1)-9255116/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+1)-9255116/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+1)-9255116/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+1)-9255116/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+1)-9255116/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+1)-9255116/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+1)+1427415/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n-1)+1427415/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n-1)+1427415/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n-1)+1427415/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n-1)+1427415/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n-1)+1427415/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n-1)+1427415/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n-1)+17895332/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n)+17895332/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n)+17895332/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n)+17895332/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n)+17895332/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n)+17895332/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n)+17895332/56686603*RootOf(_Z^7+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n)
Explicit closed form in latex syntax:
-\frac{2573401 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{56686603}-\frac{2573401 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{56686603}-\frac{2573401 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{56686603}-\frac{2573401 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{56686603}-\frac{2573401 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{56686603}-\frac{2573401 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{56686603}-\frac{2573401 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{56686603}-\frac{50727 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{56686603}-\frac{50727 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{56686603}-\frac{50727 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{56686603}-\frac{50727 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{56686603}-\frac{50727 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{56686603}-\frac{50727 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{56686603}-\frac{50727 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{56686603}+\frac{3588996 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{56686603}+\frac{3588996 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{56686603}+\frac{3588996 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{56686603}+\frac{3588996 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{56686603}+\frac{3588996 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{56686603}+\frac{3588996 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{56686603}+\frac{3588996 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{56686603}+\frac{1780405 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{56686603}+\frac{1780405 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{56686603}+\frac{1780405 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{56686603}+\frac{1780405 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{56686603}+\frac{1780405 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{56686603}+\frac{1780405 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{56686603}+\frac{1780405 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{56686603}-\frac{9255116 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{56686603}-\frac{9255116 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{56686603}-\frac{9255116 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{56686603}-\frac{9255116 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{56686603}-\frac{9255116 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{56686603}-\frac{9255116 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{56686603}-\frac{9255116 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{56686603}+\frac{1427415 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{56686603}+\frac{1427415 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{56686603}+\frac{1427415 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{56686603}+\frac{1427415 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{56686603}+\frac{1427415 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{56686603}+\frac{1427415 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{56686603}+\frac{1427415 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{56686603}+\frac{17895332 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{56686603}+\frac{17895332 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{56686603}+\frac{17895332 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{56686603}+\frac{17895332 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{56686603}+\frac{17895332 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{56686603}+\frac{17895332 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{56686603}+\frac{17895332 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{56686603}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 151
a(n) = -a(n+3)-4*a(n+4)+5*a(n+5)-4*a(n+6)+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) = 151
a \! \left(n \right) = -a \! \left(n +3\right)-4 a \! \left(n +4\right)+5 a \! \left(n +5\right)-4 a \! \left(n +6\right)+a \! \left(n +7\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19094/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[17,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[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[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[44,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[28,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[7,x]
F[27,x] = F[23,x]
F[28,x] = F[29,x]+F[39,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[2,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[32,x]*F[4,x]
F[35,x] = F[29,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[33,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[29,x]
F[43,x] = F[40,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[57,x]
F[46,x] = F[2,x]+F[47,x]
F[47,x] = F[19,x]+F[48,x]+F[56,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[11,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[26,x]*F[4,x]
F[53,x] = F[33,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[42,x]
F[56,x] = F[4,x]*F[46,x]
F[57,x] = F[29,x]+F[58,x]
F[58,x] = F[37,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{17}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{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_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{44}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{28}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{7}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{39}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{29}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{33}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{29}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{57}\! \left(x \right)
F_{46}\! \left(x \right) = F_{2}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{56}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{11}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{53}\! \left(x \right) = F_{33}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{57}\! \left(x \right) = F_{29}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{37}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_17(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_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_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_44(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_28(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_7(x))
Eq(F_27(x), F_23(x))
Eq(F_28(x), F_29(x) + F_39(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_2(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_32(x)*F_4(x))
Eq(F_35(x), F_29(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_33(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_29(x))
Eq(F_43(x), F_40(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_57(x))
Eq(F_46(x), F_2(x) + F_47(x))
Eq(F_47(x), F_19(x) + F_48(x) + F_56(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_11(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_26(x)*F_4(x))
Eq(F_53(x), F_33(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_42(x))
Eq(F_56(x), F_4(x)*F_46(x))
Eq(F_57(x), F_29(x) + F_58(x))
Eq(F_58(x), F_37(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"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, 0], [0, 1]]}, {"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, 0], [1, 0], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"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, 0], [1, 2]]}, {"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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, 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