A271359 a(n) = k*Fibonacci(2*n+1) + (k+1)*Fibonacci(2*n), where k=5.

5, 16, 43, 113, 296, 775, 2029, 5312, 13907, 36409, 95320, 249551, 653333, 1710448, 4478011, 11723585, 30692744, 80354647, 210371197, 550758944, 1441905635, 3774957961, 9882968248, 25873946783, 67738872101, 177342669520, 464289136459, 1215524739857

Offset: 0

Author

Colin Barker, Apr 05 2016

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-1).

Crossrefs

Cf. A000045.

Cf. A001906 (k=0), A002878 (k=1), A100545 (k=2, without the initial 2), A271357 (k=3), A271358 (k=4), this sequence (k=5).

Programs

Magma

k:=5; [k*Fibonacci(2*n+1)+(k+1)*Fibonacci(2*n): n in [0..30]]; // Bruno Berselli, Apr 06 2016

PARI

a(n) = 5*fibonacci(2*n+1) + 6*fibonacci(2*n)

PARI

Vec((5+x)/(1-3*x+x^2) + O(x^50))

Formula

G.f.: (5+x) / (1-3*x+x^2).

a(n) = 3*a(n-1)-a(n-2) for n>1.

a(n) = (2^(-2-n)*((13-sqrt(5))*(3+sqrt(5))^(n+1) - (13+sqrt(5))*(3-sqrt(5))^(n+1))) / sqrt(5).

a(n) = 6*Fibonacci(2*n+2) - Fibonacci(2*n+1) = 5*A001906(n+1) +A001906(n).

Extensions

Changed offset and adapted definition, programs and formulas by Bruno Berselli, Apr 06 2016