A122573 Expansion of x*(1 + x)*(1 - 3*x^2)/(1 - 4*x^2 + x^4).

1, 1, 1, 1, 3, 3, 11, 11, 41, 41, 153, 153, 571, 571, 2131, 2131, 7953, 7953, 29681, 29681, 110771, 110771, 413403, 413403, 1542841, 1542841, 5757961, 5757961, 21489003, 21489003, 80198051, 80198051, 299303201, 299303201, 1117014753

Offset

1,5

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Formula

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

a(n) = 3*b(n) + b(n-1) - 11*b(n-2) - 3*b(n-3), where a(0) = a(1) = 1, b(n) = (1/2)*(1 + (-1)^n)*c((n+2)/2), and c(n) = ((2+sqrt(3))^n - (2-sqrt(3))^n)/(2*sqrt(3)) (A001353). - G. C. Greubel, Jul 10 2021

Prog

(Magma) [n le 4 select 1 else 4*Self(n-2) - Self(n-4): n in [1..41]]; // G. C. Greubel, Jul 10 2021
(Sage)
def a(n): return 1 if (n<5) else 4*a(n-2) - a(n-4)
[a(n) for n in (1..40)] # G. C. Greubel, Jul 10 2021

Crossrefs

Cf. A001353, A001835.

Sequence in context: A146828 A146583 A146458 * A230011 A136123 A045495

Adjacent sequences: A122570 A122571 A122572 * A122574 A122575 A122576

Keyword

nonn

Author

Roger L. Bagula, Sep 17 2006

Extensions

Edited by G. C. Greubel, Jul 10 2021

Status

approved