A127211 a(n) = 4^n*Lucas(n), where Lucas = A000032.

2, 4, 48, 256, 1792, 11264, 73728, 475136, 3080192, 19922944, 128974848, 834666496, 5402263552, 34963718144, 226291089408, 1464583847936, 9478992822272, 61349312856064, 397061136580608, 2569833552019456, 16632312393367552, 107646586405781504, 696703343917006848

Offset: 0

Author

Artur Jasinski, Jan 09 2007

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1225
• Index entries for linear recurrences with constant coefficients, signature (4,16).

Crossrefs

Cf. A000032, A000204, A087131, A087131, A127210, A127212, A127213, A127214, A127215, A127216.

Programs

Magma

[4^n*Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 18 2017

Maple

a:= n-> 4^n*(<<1|1>, <1|0>>^n. <<2, -1>>)[1, 1]:
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 15 2024

Mathematica

Table[4^n Tr[MatrixPower[{{1, 1}, {1, 0}}, n]], {n, 0, 20}]
Table[4^n*LucasL[n], {n, 0, 50}] (* G. C. Greubel, Dec 18 2017 *)

PARI

my(x='x + O('x^30)); Vec(-4*x*(8*x+1)/(16*x^2+4*x-1)) \\ G. C. Greubel, Dec 18 2017

Formula

a(n) = Trace of matrix [({4,4},{4,0})^n].

a(n) = 4^n * Trace of matrix [({1,1},{1,0})^n].

From Colin Barker, Sep 02 2013: (Start)

a(n) = 4*a(n-1) + 16*a(n-2).

G.f.: 2*x*(2*x-1)/(16*x^2+4*x-1). (End)

From Peter Luschny, Apr 15 2024: (Start)

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

a(n) = 4^n*(Fibonacci(n+1) + Fibonacci(n-1)). (End)

a(n) = 2^n*A087131(n). - Michel Marcus, Apr 15 2024

Extensions

a(0)=2 prepended by Alois P. Heinz, Apr 15 2024