A141448 Generalized Pell numbers P(n,5,5).
0, 1, 2, 5, 13, 34, 89, 232, 605, 1578, 4116, 10736, 28003, 73041, 190515, 496926, 1296147, 3380779, 8818187, 23000741, 59993521, 156482896, 408159020, 1064613385, 2776862948, 7242974718, 18892067685, 49276745441, 128530009618
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
- E. Kilic and D. Tasci, The generalized Binet formula, representation and sums of the generalizedorder-k Pell numbers, Taiwanese J of Math vol 10 no 6 (2006), 1661-1670.
- Index entries for linear recurrences with constant coefficients, signature (2,1,1,1,1).
Programs
-
Magma
I:=[0,1,2,5,13]; [n le 5 select I[n] else 2*Self(n-1)+Self(n-2)+Self(n-3)+Self(n-4)+Self(n-5): n in [1..30]]; // Vincenzo Librandi, Dec 13 2012
-
Maple
P := proc(n,k,i) option remember ; if n = 1-i then 1; elif n <= 0 then 0; else 2*P(n-1,k,i)+add(P(n-j,k,i),j=2..k) ; fi ; end: for n from 0 to 40 do printf("%d,",P(n,5,5)) ; od: -
Mathematica
CoefficientList[Series[x/(1 - 2*x - x^2 - x^3 - x^4 - x^5), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 13 2012 *) LinearRecurrence[{2,1,1,1,1},{0,1,2,5,13},40] (* Harvey P. Dale, Jan 08 2016 *) -
Maxima
a(n):=b(n+1); b(n):=sum(sum(binomial(k,r)*2^(k-r)*sum((sum(binomial(j,-r+n-m-k-j)*binomial(m,j),j,0,m))*binomial(r,m),m,0,r),r,0,k),k,1,n); /* Vladimir Kruchinin, May 05 2011 */
Formula
From R. J. Mathar, Nov 28 2008: (Start)
a(n) = 2*a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5).
G.f.: x/(1-2*x-x^2-x^3-x^4-x^5). (End)
a(n+1) = Sum_(k=1..n, Sum_(r=0..k, binomial(k,r)*2^(k-r)*Sum_(m=0..r,(Sum_(j=0..m, binomial(j,-r+n-m-k-j)*binomial(m,j)))*binomial(r,m)))), a(0)=0, a(1)=1. [Vladimir Kruchinin, May 05 2011]
Comments