A074662 a(n) = Fibonacci(n+1)+cos(n*Pi/2).

2, 1, 1, 3, 6, 8, 12, 21, 35, 55, 88, 144, 234, 377, 609, 987, 1598, 2584, 4180, 6765, 10947, 17711, 28656, 46368, 75026, 121393, 196417, 317811, 514230, 832040, 1346268, 2178309, 3524579, 5702887, 9227464, 14930352, 24157818, 39088169

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 29 2002

Keywords

• easy
• nonn

Comments

a(n) is the convolution of L(n) with the sequence (1,0,-1,0,1,0,-1,0,....) A056594. a(2n+1)=F(2n+2), F = Fibonacci numbers.

a(n) = Sum((-1)^(i+floor(n/2))L(2i+e),(i=0..floor(n/2))), where L(n) Lucas numbers and e=(1/2)(1-(-1)^n).

Examples

			G.f. = 2 + x + x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 12*x^6 + 21*x^7 + 35*x^8 + ...
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Crossrefs

Cf. A000045, A056594.

Programs

Mathematica

CoefficientList[Series[(2 - x)/(1 - x - x^3 - x^4), {x, 0, 40}], x]
a[ n_] := Fibonacci[n + 1] + Re[I^n] (* Michael Somos, Jan 19 2014 *)
LinearRecurrence[{1,0,1,1},{2,1,1,3},50] (* Harvey P. Dale, Sep 08 2023 *)

PARI

{a(n) = fibonacci(n+1) + real(I^n)} /* Michael Somos, Jan 19 2014 */

Formula

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

G.f.: (2 - x)/(1 - x - x^3 - x^4).

a(4n) = F(4n+1)+1, a(4n+2) = F(4n+3)-1.

a(n+1)*a(n+3) = a(n)*a(n+2) + a(n+1)*a(n+2) for all n in Z. - Michael Somos, Jan 19 2014