A271357 a(n) = k*Fibonacci(2*n+1) + (k+1)*Fibonacci(2*n), where k=3.

3, 10, 27, 71, 186, 487, 1275, 3338, 8739, 22879, 59898, 156815, 410547, 1074826, 2813931, 7366967, 19286970, 50493943, 132194859, 346090634, 906077043, 2372140495, 6210344442, 16258892831, 42566334051, 111440109322, 291753993915, 763821872423, 1999711623354

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), this sequence (k=3), A271358 (k=4), A271359 (k=5).

Programs

Magma

k:=3; [k*Fibonacci(2*n+1)+(k+1)*Fibonacci(2*n): n in [0..30]]; // Bruno Berselli, Apr 06 2016

Mathematica

Table[3Fibonacci[2n+1]+4Fibonacci[2n],{n,0,30}] (* or *) LinearRecurrence[ {3,-1},{3,10},30] (* Harvey P. Dale, Apr 05 2019 *)

PARI

a(n) = 3*fibonacci(2*n+1) + 4*fibonacci(2*n)

PARI

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

Formula

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

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

a(n) = (2^(-2-n)*((9-sqrt(5))*(3+sqrt(5))^(n+1) - (9+sqrt(5))*(3-sqrt(5))^(n+1))) / sqrt(5).

a(n) = 4*Fibonacci(2*n+2) - Fibonacci(2*n+1).

Extensions

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