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

0,3

Comments

P(n,2,2) and P(n,2,1) are in A000129.

P(n,3,2) is A116413. P(n,3,1) and P(n,3,3) are A077939.

P(n,4,1) and P(n,4,4) are A103142.

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).

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]

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 *)

Prog

(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 */
(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

Crossrefs

Sequence in context: A103142 A112844 A027933 * A011783 A001519 A048575

Adjacent sequences: A141445 A141446 A141447 * A141449 A141450 A141451

Keyword

nonn,easy

Author

R. J. Mathar, Aug 07 2008

Status

approved