A082762 Trinomial transform of Lucas numbers (A000032).

1, 8, 44, 232, 1216, 6368, 33344, 174592, 914176, 4786688, 25063424, 131233792, 687149056, 3597959168, 18839158784, 98643116032, 516502061056, 2704439902208, 14160631169024, 74146027405312, 388233639755776, 2032817728913408, 10643971814457344

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Yuhan Jiang, The doubly asymmetric simple exclusion process, the colored Boolean process, and the restricted random growth model, arXiv:2312.09427 [math.CO], 2023.

Index entries for linear recurrences with constant coefficients, signature (6,-4).

Formula

a(n) = Sum_{k=0..2*n} Trinomial(n,k)*Lucas(k+1), where Trinomial(n,k) = trinomial coefficients (A027907).

a(n) = 2^n*Lucas(2*n+1), where Lucas = A000032.

From Philippe Deléham, Mar 01 2004: (Start)

a(n) = 2^n*A002878(n) = 2^(-n)*Sum_{k>=0} C(2*n+1,2*k)*5^k; see A091042.

a(0) = 1, a(1) = 8, a(n+1) = 6*a(n) - 4*a(n-1). (End)

From Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009: (Start)

a(n) = ((1+sqrt(5))*(3+sqrt(5))^n + (1-sqrt(5))*(3-sqrt(5))^n)/2.

Third binomial transform of 1, 5, 5, 25, 25, 125. (End)

G.f.: (1 + 2*x)/(1 - 6*x + 4*x^2). - Colin Barker, Mar 23 2012

Mathematica

a[n_]:=(MatrixPower[{{2, 2}, {2, 4}}, n].{{2}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(3 + s)^n + (1 - s)(3 - s)^n)/2]]; Array[f, 21, 0] (* Robert G. Wilson v, Mar 07 2011 *)
LinearRecurrence[{6, -4}, {1, 8}, 30] (* G. C. Greubel, Dec 21 2017 *)

Prog

(PARI) x='x+O('x^30); Vec((1 + 2*x)/(1 - 6*x + 4*x^2)) \\ G. C. Greubel, Dec 21 2017
(Magma) I:=[1, 8]; [n le 2 select I[n] else 6*Self(n-1)-4*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 21 2017

Crossrefs

Cf. A000032, A027907, A091042, A292277.

Sequence in context: A270935 A197213 A198768 * A249794 A322498 A147828

Adjacent sequences: A082759 A082760 A082761 * A082763 A082764 A082765

Keyword

nonn,easy

Author

Emanuele Munarini, May 21 2003

Status

approved