A074331 a(n) = Fibonacci(n+1) - (1 + (-1)^n)/2.

0, 1, 1, 3, 4, 8, 12, 21, 33, 55, 88, 144, 232, 377, 609, 987, 1596, 2584, 4180, 6765, 10945, 17711, 28656, 46368, 75024, 121393, 196417, 317811, 514228, 832040, 1346268, 2178309, 3524577, 5702887, 9227464, 14930352, 24157816, 39088169

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 21 2002

Keywords

• nonn
• easy

Comments

a(n) is the convolution of F(n) with the sequence (1,0,1,0,1,0,...).

Transform of F(n) under the Riordan array (1/(1-x^2), x). - Paul Barry, Apr 16 2005

Links

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

Crossrefs

Essentially the same as A052952.

Cf. A000045.

Programs

Magma

[Fibonacci(n+1) - (1+(-1)^n)/2: n in [0..40]]; // G. C. Greubel, Jun 23 2022

Maple

with(combinat):seq(fibonacci(n+1)-(1+(-1)^n)/2, n=0..40); # Zerinvary Lajos, Mar 17 2008

Mathematica

CoefficientList[Series[x/(1-x-2*x^2+x^3+x^4), {x, 0, 40}], x]
Table[Floor[GoldenRatio^(k+1)/Sqrt[5]], {k, 0, 40}] (* Federico Provvedi, Mar 26 2013 *)

PARI

a(n)=if(n<0,0,fibonacci(n+1)-(n+1)%2)

SageMath

[fibonacci(n+1) -((n+1)%2) for n in (0..40)] # G. C. Greubel, Jun 23 2022

Formula

a(n) = Sum_{i=0..floor(n/2)} Fibonacci(2*i + e), where e = 2*(n/2 - floor(n/2)).

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n > 3, a(0)=0, a(1)=1, a(2)=1, a(3)=3.

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

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

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

a(n-1) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1). - Paul Barry, Jul 07 2004

a(n) = Sum_{k=0..floor((n-1)/2)} Fibonacci(n-2*k). - Paul Barry, Apr 16 2005

a(n) = Sum_{k=0..n} Fibonacci(k)*(1-(-1)^floor((n+k-1)/2)). - Paul Barry, Apr 16 2005

a(n) = Fibonacci(n) + a(n-2) for n > 1. - Zerinvary Lajos, Mar 17 2008

a(n) = floor(g^(n+1)/sqrt(5)), where g = (sqrt(5) + 1)/2. - Federico Provvedi, Mar 27 2013

E.g.f.: exp(x/2)*(cosh(sqrt(5)*x/2) + (1/sqrt(5))*sinh(sqrt(5)*x/2)) - cosh(x). - G. C. Greubel, Jun 23 2022