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A127211
a(n) = 4^n*Lucas(n), where Lucas = A000032.
11
2, 4, 48, 256, 1792, 11264, 73728, 475136, 3080192, 19922944, 128974848, 834666496, 5402263552, 34963718144, 226291089408, 1464583847936, 9478992822272, 61349312856064, 397061136580608, 2569833552019456, 16632312393367552, 107646586405781504, 696703343917006848
OFFSET
0,1
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
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 *)
PROG
(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
(Magma) [4^n*Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 18 2017
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Jan 09 2007
EXTENSIONS
a(0)=2 prepended by Alois P. Heinz, Apr 15 2024
STATUS
approved