0123_0231_1203_1302_2031
Counting sequence:
1, 1, 2, 6, 19, 57, 163, 457, 1280, 3602, 10171, 28749, 81241, 229477, 648046, 1830030, 5168055, 14595229, 41219259, 116409685, 328758020, 928458634, 2622095607, 7405162713, 20913216433, 59061860329, 166798989978, 471063762294, 1330350158043, 3757095474817, 10610564702739, 29965723365825, 84627407065992, 238999671095778, 674968603543987, 1906206036057125, 5383393290419593, 15203457953258381, 42936698337493510, 121259260217291070, 342453163793137535, 967135781478154085, 2731327138148930091, 7713651049265121805, 21784432805126503436, 61522294644945467034, 173747591789006264303, 490687576377014213809, 1385770560222409505121, 3913610488690266082385, 11052585108120699464242, 31214051047050751748710, 88152859556089644005283, 248956043424229360724169, 703086795704124941001347, 1985615755674335698754873, 5607657480231400091012624, 15836811490708505197114162, 44725377588840800992971179, 126310741378591176854678781, 356719255324745357536397753, 1007425225524079289426290485, 2845110180829194923214182302, 8034990325805036499490811758, 22691940006683496897608517959, 64085222307387417790810450989, 180985659092067439890776642715, 511128893957415996839807563333, 1443499709030714331106007349012, 4076645704449955886513935624490, 11513019431621279522292360574183, 32514382176553043062161802344329, 91825177105088575592324972690705, 259327183416737852774364087997753, 732376350136455588419988760773322, 2068333566779396597797398719644374, 5841264184280810684461558427513963, 16496549598472459355180335216427793, 46588570567856941083438636516990451, 131572659761358133175751984086567985, 371579651096267294868847932164878488, 1049393068128689178132340085476425218, 2963633256524165071528701320342788003, 8369716120612810949828259906167305941, 23637252613977327957143256604631197161, 66754917739799026641492083426470927197, 188525253557270674105171682970627676214, 532421766548643814588762895058421680926, 1503633768664655072849096569486971377935, 4246472725795497347925828944969929262837, 11992634767001372419612118689507286939659, 33868883174740620852708885065985176367613, 95650477963404907214409290815687170293788, 270130369738649463203247166424712030206778, 762886064020060118850021653252690274142175, 2154497279365879100212052849273779389330785, 6084602597581225499448774877536016416721601, 17183771418541213182590261415031831126583969, 48529381406446417426473991027465663594182754, 137053781869514620802310439405297337688302278, 387059108943048468418464358731247926372288307
Generating function in Maple syntax:
-(x-1)^4/(x^5-3*x^4+9*x^3-9*x^2+5*x-1)
Generating function in latex syntax:
-\frac{\left(x -1\right)^{4}}{x^{5}-3 x^{4}+9 x^{3}-9 x^{2}+5 x -1}
Generating function in sympy syntax:
-(x - 1)**4/(x**5 - 3*x**4 + 9*x**3 - 9*x**2 + 5*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^5-3*x^4+9*x^3-9*x^2+5*x-1)*F(x)+(x-1)^4 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}-3 x^{4}+9 x^{3}-9 x^{2}+5 x -1\right) F \! \left(x \right)+\left(x -1\right)^{4} = 0
Explicit closed form in Maple syntax:
19/2563*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n+3)+19/2563*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n+3)+19/2563*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n+3)+19/2563*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n+3)+19/2563*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n+3)-9/233*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n+2)-9/233*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n+2)-9/233*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n+2)-9/233*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n+2)-9/233*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n+2)+375/5126*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n+1)+375/5126*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n+1)+375/5126*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n+1)+375/5126*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n+1)+375/5126*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n+1)+471/5126*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n-1)+471/5126*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n-1)+471/5126*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n-1)+471/5126*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n-1)+471/5126*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n-1)+89/2563*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n)+89/2563*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n)+89/2563*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n)+89/2563*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n)+89/2563*RootOf(_Z^5-3*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n)
Explicit closed form in latex syntax:
\frac{19 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{2563}+\frac{19 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{2563}+\frac{19 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{2563}+\frac{19 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{2563}+\frac{19 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{2563}-\frac{9 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{233}-\frac{9 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{233}-\frac{9 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{233}-\frac{9 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{233}-\frac{9 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{233}+\frac{375 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{5126}+\frac{375 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{5126}+\frac{375 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{5126}+\frac{375 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{5126}+\frac{375 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{5126}+\frac{471 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{5126}+\frac{471 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{5126}+\frac{471 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{5126}+\frac{471 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{5126}+\frac{471 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{5126}+\frac{89 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{2563}+\frac{89 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{2563}+\frac{89 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{2563}+\frac{89 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{2563}+\frac{89 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{2563}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+5) = a(n)-3*a(n+1)+9*a(n+2)-9*a(n+3)+5*a(n+4), n >= 5
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(n +5\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+9 a \! \left(n +2\right)-9 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17318/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[19,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[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[17,x]+F[18,x]
F[17,x] = F[11,x]
F[18,x] = F[14,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[40,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[15,x]
F[26,x] = F[20,x]+F[27,x]
F[27,x] = 2*F[21,x]+F[28,x]+F[32,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[25,x]
F[31,x] = F[27,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[38,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[2,x]+F[35,x]
F[38,x] = F[39,x]
F[39,x] = F[32,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[37,x]+F[42,x]
F[42,x] = F[39,x]+F[43,x]
F[43,x] = F[40,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{19}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{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_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{40}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \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_{38}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{32}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{39}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_19(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_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_17(x) + F_18(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_40(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_15(x))
Eq(F_26(x), F_20(x) + F_27(x))
Eq(F_27(x), 2*F_21(x) + F_28(x) + F_32(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_25(x))
Eq(F_31(x), F_27(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_38(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_2(x) + F_35(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_32(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_37(x) + F_42(x))
Eq(F_42(x), F_39(x) + F_43(x))
Eq(F_43(x), F_40(x))
Pack JSON:
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Specification JSON:
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