A116698 Expansion of (1-x+3*x^2+x^3) / ((1-x-x^2)*(1+2*x^2)).

1, 0, 2, 5, 5, 4, 13, 29, 34, 39, 89, 176, 233, 313, 610, 1115, 1597, 2328, 4181, 7277, 10946, 16687, 28657, 48416, 75025, 117297, 196418, 326003, 514229, 815656, 1346269, 2211077, 3524578, 5637351, 9227465

Offset: 0

Author

Creighton Dement, Feb 23 2006

Keywords

• easy
• nonn
• changed

Links

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

Crossrefs

Cf. A000045, A001519, A006498, A115008, A116697, A116699.

Programs

Magma

A116698:= func< n | Fibonacci(n+1) -((n mod 2) -2*0^((n+1) mod 4))*2^Floor(n/2) >;
[A116898(n): n in [0..50]]; // G. C. Greubel, Aug 24 2025

Mathematica

CoefficientList[Series[(1-x+3x^2+x^3)/((1-x-x^2)(1+2x^2)),{x,0,100}],x] (* or *) LinearRecurrence[{1,-1,2,2},{1,0,2,5},100] (* Harvey P. Dale, May 14 2022 *)
Table[Fibonacci[n+1] -I^(n-1)*Mod[n,2]*2^Floor[n/2], {n,0,50}] (* G. C. Greubel, Aug 24 2025 *)

PARI

Vec((1-x +3*x^2 +x^3)/((1-x-x^2)*(1+2*x^2)) + O(x^40)) \\ Colin Barker, May 18 2019

SageMath

def A116898(n): return fibonacci(n+1) - (-1)**((n-1)//2)*(n%2)*2**(n//2)
print([A116898(n) for n in range(51)]) # G. C. Greubel, Aug 24 2025

Formula

a(2*n) = A000045(2*n+1) = A001519(n).

a(n) = a(n-1) - a(n-2) + 2*a(n-3) + 2*a(n-4) for n > 3. - Colin Barker, May 18 2019

From G. C. Greubel, Aug 24 2025: (Start)

a(n) = A000045(n+1) - (-1)^floor((n-1)/2) * (n mod 2) * 2^floor(n/2).

E.g.f.: exp(x/2)*(cosh(sqrt(5)*x/2) + (1/sqrt(5))*sinh(sqrt(5)*x/2)) - sin(sqrt(2)*x)/sqrt(2). (End)