A083882 a(0)=1, a(1)=4, a(n)=8a(n-1)-13a(n-2), n>=2.

1, 4, 19, 100, 553, 3124, 17803, 101812, 583057, 3340900, 19147459, 109747972, 629066809, 3605810836, 20668618171, 118473404500, 679095199777, 3892607339716, 22312621120627, 127897073548708, 733112513821513

Offset: 0

Author

Paul Barry, May 08 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A083881.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1319
• Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
• Index entries for linear recurrences with constant coefficients, signature (8,-13).

Programs

Mathematica

LinearRecurrence[{8,-13},{1,4},30] (* Harvey P. Dale, Aug 02 2015 *)

Formula

a(n) = ((4-sqrt(3))^n+(4+sqrt(3))^n)/2.

a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)4^(n-2k)3^k.

G.f.: (1-4x)/(1-8x+13x^2);

E.g.f.: exp(4x)cosh(x*sqrt(3)).

a(n) = Sum_{k=0..n} A027907(n,2k)*3^k . - J. Conrad, Aug 24 2016