A121401 a(n) = ((sqrt(3)+1)^n+(sqrt(3)-1)^n)^2/2^(n+1).
2, 3, 8, 27, 98, 363, 1352, 5043, 18818, 70227, 262088, 978123, 3650402, 13623483, 50843528, 189750627, 708158978, 2642885283, 9863382152, 36810643323, 137379191138, 512706121227, 1913445293768, 7141075053843, 26650854921602, 99462344632563, 371198523608648
Offset: 0
Links
- Stefano Spezia, Table of n, a(n) for n = 0..1700
- Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 53, 56.
- Index entries for linear recurrences with constant coefficients, signature (5,-5,1).
Programs
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Mathematica
Table[((-1+Sqrt[3])^n+(1+Sqrt[3])^n)^2/(2^(n+1)),{n,0,25}] LinearRecurrence[{5,-5,1},{2,3,8},25] (* Ray Chandler, Jan 27 2014 *)
Formula
a(n) = ((2+sqrt(3))^n+(2-sqrt(3))^n)/2.
a(n) = A001075(n)+1.
From R. J. Mathar, Aug 07 2008: (Start)
a(n) = A102206(n-1).
G.f.: (1-3*x)*(x-2)/((x-1)*(x^2-4*x+1)). (End)
a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3). - Wesley Ivan Hurt, Jan 16 2024
E.g.f.: cosh(x) + exp(2*x)*cosh(sqrt(3)*x) + sinh(x). - Stefano Spezia, Jul 09 2025