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

Author

Emanuele Munarini, May 21 2003

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000032, A027907, A091042, A292277.

Programs

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

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 *)

PARI

x='x+O('x^30); Vec((1 + 2*x)/(1 - 6*x + 4*x^2)) \\ G. C. Greubel, Dec 21 2017

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