OFFSET
1,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,9,27).
FORMULA
a(n) = Trace of matrix [({3,3,3},{3,0,0},{0,3,0})^n].
a(n) = 3^n * Trace of matrix [({1,1,1},{1,0,0},{0,1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 3*a(n-1) + 9*a(n-2) + 27*a(n-3).
G.f.: -3*x*(27*x^2+6*x+1)/(27*x^3+9*x^2+3*x-1). (End)
MATHEMATICA
Table[Tr[MatrixPower[3*{{1, 1, 1}, {1, 0, 0}, {0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{3, 9, 27}, {3, 27, 189}, 50] (* G. C. Greubel, Dec 18 2017 *)
PROG
(PARI) x='x+O('x^30); Vec(-3*x*(27*x^2+6*x+1)/(27*x^3+9*x^2+3*x-1)) \\ G. C. Greubel, Dec 18 2017
(Magma) I:=[3, 27, 189]; [n le 3 select I[n] else 3*Self(n-1) + 9*Self(n-2) + 27*Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 18 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Jan 09 2007
EXTENSIONS
More terms from Colin Barker, Sep 02 2013
STATUS
approved