A117719 a(2n) = A001653(n) (Numbers n such that 2*n^2 - 1 is a square), a(2n+1) = A038725(n+1).

1, 2, 5, 11, 29, 64, 169, 373, 985, 2174, 5741, 12671, 33461, 73852, 195025, 430441, 1136689, 2508794, 6625109, 14622323, 38613965, 85225144, 225058681, 496728541, 1311738121, 2895146102, 7645370045, 16874148071, 44560482149

Offset: 0

Author

Creighton Dement, Apr 13 2006

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Crossrefs

Cf. A000129, A001653, A038725.

Programs

Magma

I:= [1,2,5,11]; [n le 4 select I[n] else 6*Self(n-2) -Self(n-4): n in [1..40]]; // G. C. Greubel, Jul 23 2023

Mathematica

LinearRecurrence[{0,6,0,-1}, {1,2,5,11}, 40] (* G. C. Greubel, Jul 23 2023 *)

SageMath

A000129=BinaryRecurrenceSequence(2,1,0,1)
def A117719(n): return (3*A000129(n+1) +2*A000129(n) +(-1)^n*A000129(n-1))/4
[A117719(n) for n in range(41)] # G. C. Greubel, Jul 23 2023

Formula

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

a(n) = (1/4)*( 3*P(n+1) + 2*P(n) + (-1)^n*P(n-1) ), where P(n) = A000129(n). - G. C. Greubel, Jul 23 2023