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A127210
a(n) = 3^n*Lucas(n), where Lucas = A000204.
10
3, 27, 108, 567, 2673, 13122, 63423, 308367, 1495908, 7263027, 35252253, 171124002, 830642283, 4032042867, 19571909148, 95004113247, 461159522073, 2238515585442, 10865982454983, 52744587633927, 256027604996628, 1242784103695227, 6032600756055333, 29282859201423042
OFFSET
1,1
LINKS
Ivica Martinjak, Two Extensions of the Sury's Identity, arXiv:1508.01444 [math.CO], 2015.
FORMULA
a(n) = Trace of matrix [({3,3},{3,0})^n] = 3^n * Trace of matrix [({1,1},{1,0})^n].
From R. J. Mathar, Oct 27 2008: (Start)
a(n) = 3*a(n-1) + 9*a(n-2).
G.f.: 3*x*(1 + 6*x)/(1 - 3*x - 9*x^2).
a(n) = 3*A099012(n) +18*A099012(n-1). (End)
MATHEMATICA
Table[3^n Tr[MatrixPower[{{1, 1}, {1, 0}}, x]], {x, 1, 20}]
Table[3^n LucasL[n], {n, 25}] (* Vincenzo Librandi, Aug 07 2015 *)
PROG
(PARI) lucas(n) = fibonacci(n-1) + fibonacci(n+1);
vector(30, n, 3^n*lucas(n)) \\ Michel Marcus, Aug 07 2015
(Magma) [3^n*Lucas(n): n in [1..30]]; // Vincenzo Librandi, Aug 07 2015
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Jan 09 2007
EXTENSIONS
More terms from Michel Marcus, Aug 07 2015
STATUS
approved