OFFSET
1,1
LINKS
Indranil Ghosh, Table of n, a(n) for n = 1..1702
Index entries for linear recurrences with constant coefficients, signature (2,4,8,16).
FORMULA
a(n) = Trace of matrix [({2,2,2,2},{2,0,0,0},{0,2,0,0},{0,0,2,0})^n].
a(n) = 2^n * Trace of matrix [({1,1,1,1},{1,0,0,0},{0,1,0,0},{0,0,1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 2*a(n-1) + 4*a(n-2) + 8*a(n-3) + 16*a(n-4).
G.f.: -2*x*(32*x^3+12*x^2+4*x+1) / (16*x^4+8*x^3+4*x^2+2*x-1). (End)
EXAMPLE
a(8) = (2^8) * A001648(8) = 256 * 191 = 48896. - Indranil Ghosh, Feb 09 2017
MATHEMATICA
Table[Tr[MatrixPower[2*{{1, 1, 1, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{2, 4, 8, 16}, {2, 12, 56, 240}, 50] (* G. C. Greubel, Dec 19 2017 *)
PROG
(PARI) x='x+O('x^30); Vec(-2*x*(32*x^3+12*x^2+4*x+1)/(16*x^4 +8*x^3 +4*x^2 +2*x -1)) \\ G. C. Greubel, Dec 19 2017
(Magma) I:=[2, 12, 56, 240]; [n le 4 select I[n] else 2*Self(n-1) + 4*Self(n-2) + 8*Self(n-3) + 16*Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 19 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Jan 09 2007
EXTENSIONS
More terms from Colin Barker, Sep 02 2013
STATUS
approved