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

Author

Creighton Dement, Jul 18 2005

Keywords

• easy
• nonn

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).

Crossrefs

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

Programs

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

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 *)

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

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

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