A127210 a(n) = 3^n*Lucas(n), where Lucas = A000204.

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

Author

Artur Jasinski, Jan 09 2007

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1445
• Ivica Martinjak, Two Extensions of the Sury's Identity, arXiv:1508.01444 [math.CO], 2015.
• Index entries for linear recurrences with constant coefficients, signature (3, 9).

Crossrefs

Cf. A000204, A087131, A127211, A127212, A127213, A127214, A127215, A127216.

Programs

Magma

[3^n*Lucas(n): n in [1..30]]; // Vincenzo Librandi, Aug 07 2015

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 *)

PARI

lucas(n) = fibonacci(n-1) + fibonacci(n+1);
vector(30, n, 3^n*lucas(n)) \\ Michel Marcus, Aug 07 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)

Extensions

More terms from Michel Marcus, Aug 07 2015