A254884 a(n) = Fibonacci(2*n) + ((-1)^n-1)*Fibonacci(n).

0, -1, 3, 4, 21, 45, 144, 351, 987, 2516, 6765, 17533, 46368, 120927, 317811, 830820, 2178309, 5699693, 14930352, 39079807, 102334155, 267892404, 701408733, 1836254589, 4807526976, 12586118975, 32951280099, 86267178436, 225851433717, 591285701421, 1548008755920

Offset: 0

Author

Peter Luschny, Mar 09 2015

Keywords

• sign
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (3, 2, -9, 2, 3, -1).

Crossrefs

Cf. A000032, A000045, A005522, A006172, A022112, A033888.

Programs

Maple

gf := x -> x/(x^2-3*x+1) + x/(x^2-x-1) + x/(x^2+x-1):
seq(coeff(series(gf(x),x,n+1),x,n), n=0..30);

Mathematica

LinearRecurrence[{4,-1,-11,11,1,-4,1}, {0,-1,3,4,21,45,144}, 31]
LinearRecurrence[{3, 2, -9, 2, 3, -1},{0, -1, 3, 4, 21, 45},31] (* Ray Chandler, Aug 03 2015 *)

Sage

A254884 = lambda n: fibonacci(2*n) + ((-1)^n-1)*fibonacci(n)
[A254884(n) for n in range(31)]

Formula

Let phi = (1+sqrt(5))/2, p(n) = phi^n - (-phi)^(-n) and FL(n) = 1 + (p(n-1) + p(n+1) + p(2*n-1)) / sqrt(5).

a(n) = FL(-n) - FL(n). By this definition a(n) is a doubly infinite sequence.

a(n) = -a(-n) for all n in Z.

a(n) = A006172(n) - A005522(n).

a(2*n) = A033888(n).

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

a(n) = 4*a(n-1) - a(n-2) - 11*a(n-3) + 11*a(n-4) + a(n-5) - 4*a(n-6) + a(n-7).