A110293 a(2*n) = A001570(n), a(2*n+1) = A011943(n+1).

1, 7, 13, 97, 181, 1351, 2521, 18817, 35113, 262087, 489061, 3650401, 6811741, 50843527, 94875313, 708158977, 1321442641, 9863382151, 18405321661, 137379191137, 256353060613, 1913445293767, 3570537526921, 26650854921601, 49731172316281, 371198523608647, 692665874901013

Offset

0,2

Comments

See also A110294 (compare program code).

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 30, 56.

Index entries for linear recurrences with constant coefficients, signature (0,14,0,-1).

Formula

G.f.: (1+7*x-x^2-x^3) / ((1-4*x+x^2)*(1+4*x+x^2)).

a(2*n+1) = (a(2*n) + a(2*n+2))/2 and see A232765 for Diophantine equation that produces a sequence related to a(n). - Richard R. Forberg, Nov 30 2013

From Colin Barker, Nov 01 2016: (Start)

a(n) = (3-(-1)^n)*((-3+2*sqrt(3))*(2-sqrt(3))^n + (3+2*sqrt(3))*(2+sqrt(3))^n )/(8*sqrt(3)).

a(n) = 14*a(n-2) - a(n-4) for n>3. (End)

a(n) = (1/4)*(3 - (-1)^n)*(2*A001353(n) - A001353(n-1)). - G. C. Greubel, Jan 04 2023

Maple

seriestolist(series((1+7*x-x^2-x^3)/((1-4*x+x^2)*(1+4*x+x^2)), x=0, 25));

Mathematica

CoefficientList[Series[(1+7x-x^2-x^3)/((1-4x+x^2)(1+4x+x^2)), {x, 0, 25}], x] (* Michael De Vlieger, Nov 01 2016 *)
LinearRecurrence[{0, 14, 0, -1}, {1, 7, 13, 97}, 40] (* Harvey P. Dale, May 14 2026 *)

Prog

(PARI) Vec((1+7*x-x^2-x^3)/((1-4*x+x^2)*(1+4*x+x^2)) + O(x^30)) \\ Colin Barker, Nov 01 2016
(Magma)
A001353:= func< n | Evaluate(ChebyshevSecond(n+1), 2) >;
[(3-(-1)^n)*(2*A001353(n) - A001353(n-1))/4: n in [0..40]]; // G. C. Greubel, Jan 04 2023
(SageMath)
def A001353(n): return chebyshev_U(n, 2)
[(3-(-1)^n)*(2*A001353(n) - A001353(n-1))/4 for n in range(41)] # G. C. Greubel, Jan 04 2023

Crossrefs

Cf. A001353, A001570, A011943, A110294, A232765.

Sequence in context: A320461 A132373 A387258 * A253333 A039687 A001544

Adjacent sequences: A110290 A110291 A110292 * A110294 A110295 A110296

Keyword

easy,nonn

Author

Creighton Dement, Jul 18 2005

Status

approved