A087124 a(n) = Fibonacci(n) + Fibonacci(2n+1).

1, 3, 6, 15, 37, 94, 241, 623, 1618, 4215, 11001, 28746, 75169, 196651, 514606, 1346879, 3525565, 9229062, 24160401, 63250167, 165586906, 433505383, 1134920881, 2971243730, 7778788417, 20365086099, 53316412566, 139584058863

Offset: 0

Author

Paul Barry, Aug 15 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A087123.

For n>=1, a(n) is the coefficient of x in the reduction by x^2->x+1 of the polynomial 1+x^n+x^(2n+1). For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232. - Clark Kimberling, Jul 01 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Crossrefs

Cf. A000045, A087123, A192232.

Programs

Magma

[Fibonacci(n)+Fibonacci(2*n+1): n in [0..40]]; // Vincenzo Librandi, Mar 13 2012

Mathematica

CoefficientList[Series[(1-2*x)*(1+x-x^2)/((1-3*x+x^2)*(1-x-x^2)),{x,0,1001}],x] (* Vincenzo Librandi, Mar 13 2012 *)
LinearRecurrence[{4,-3,-2,1},{1,3,6,15},30] (* Harvey P. Dale, Aug 17 2024 *)

PARI

a(n)=fibonacci(n)+fibonacci(2*n+1) \\ Charles R Greathouse IV, Mar 13 2012

Formula

G.f.: (1-2*x)*(1+x-x^2)/((1-3*x+x^2)*(1-x-x^2)). - Colin Barker, Mar 12 2012