A279007 Values tilde(B_s(2)) of q-analogs of Fibonacci numbers.

0, 2, 1, 10, 9, 52, 65, 278, 429, 1520, 2701, 8458, 16545, 47692, 99641, 271550, 593589, 1557032, 3511045, 8972338, 20669289, 51883780, 121291121, 300757478, 710223165, 1746369632, 4152630781, 10152294490, 24255893169, 59066734012, 141584054825, 343845456398, 826053742149, 2002395391640

Offset

0,2

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

Index entries for linear recurrences with constant coefficients, signature (0,5,2).

Formula

From Thomas Scheuerle, Jan 12 2026: (Start)

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

E.g.f.: (1/14)*(-6*exp(-2*x) + exp(x)*(6*cosh(sqrt(2)*x) + 5*sqrt(2)*sinh(sqrt(2)*x))).

a(n) = 5*a(n-2) + 2*a(n-3).

a(n) = A000129(n-1) + 4*a(n-2), for n > 1. (End)

Mathematica

LinearRecurrence[{0, 5, 2}, {0, 2, 1}, 50] (* Paolo Xausa, Apr 08 2026 *)

Prog

(PARI) a(n) = (2/7)*([2, 1; 1, 0]^n)[2, 1]+(3/7)*(([2, 1; 1, 0]^(n+1))[2, 1]-(-2)^n) \\ Thomas Scheuerle, Jan 12 2026

Crossrefs

Cf. A000129.

Cf. A007179, A000124, A002856, A003600, A254875. (other q-deformations as mentioned in the link section).

Cf. Another type of q-analogs of Fibonacci numbers: A000045, A015474, A015475, A015476, A015477, A015479, A015480, A015481, A015482, A015484, A015485.

Sequence in context: A138098 A081098 A244582 * A163914 A167883 A290596

Adjacent sequences: A279004 A279005 A279006 * A279008 A279009 A279010

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 07 2016

Extensions

More terms from Thomas Scheuerle, Jan 12 2026

Status

approved