A104554 Expansion of x*(1-x)/(1-x+2*x^3-x^4).

0, 1, 0, 0, -2, -1, -1, 3, 3, 4, -3, -6, -11, -1, 8, 24, 15, -2, -42, -48, -29, 53, 107, 117, -18, -179, -306, -153, 187, 620, 620, 93, -960, -1580, -1146, 867, 3067, 3779, 899, -4368, -8859, -6878, 2757, 16107, 21004, 8612, -20845, -46746, -42966, 7336, 79983, 119169, 61531, -91099, -249454, -253347, -9618

Offset

0,5

Comments

A transform of the Fibonacci numbers A000045 under the Riordan array (1, x*(1-x)).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,-2,1).

Formula

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

a(n) = Sum_{k=0..n} binomial(k, n-k)*(-1)^(n-k)*Fibonacci(k).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*Fibonacci(n-k).

Mathematica

LinearRecurrence[{1, 0, -2, 1}, {0, 1, 0, 0}, 60] (* G. C. Greubel, Jan 01 2023 *)
CoefficientList[Series[x (1-x)/(1-x+2x^3-x^4), {x, 0, 60}], x] (* Harvey P. Dale, Mar 24 2024 *)

Prog

(Magma) I:=[0, 1, 0, 0]; [n le 4 select I[n] else Self(n-1) -2*Self(n-3) +Self(n-4): n in [1..61]]; // G. C. Greubel, Jan 01 2023
(SageMath)
@CachedFunction
def a(n): # a = A104554
    if (n<4): return (0, 1, 0, 0)[n]
    else: return a(n-1) - 2*a(n-3) + a(n-4)
[a(n) for n in range(61)] # G. C. Greubel, Jan 01 2023

Crossrefs

Cf. A000045.

Sequence in context: A260056 A269699 A035636 * A293304 A152414 A184834

Adjacent sequences: A104551 A104552 A104553 * A104555 A104556 A104557

Keyword

easy,sign

Author

Paul Barry, Mar 14 2005

Status

approved