A127220 a(n) = 3^n*tetranacci(n) or (2^n)*A001648(n).
3, 27, 189, 1215, 6318, 37179, 216513, 1253151, 7223661, 41806692, 241805655, 1398221271, 8084811933, 46753521975, 270362105694, 1563413859999, 9040715391141, 52279683047127, 302316992442837, 1748203962973380, 10109314209860523, 58458991419115875
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (3,9,27,81).
Crossrefs
Programs
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Magma
I:=[3, 27, 189, 1215]; [n le 4 select I[n] else 3*Self(n-1) + 9*Self(n-2) + 27*Self(n-3) + 81*Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 19 2017
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Mathematica
Table[Tr[MatrixPower[3*{{1, 1, 1, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}}, x]], {x, 1, 20}] LinearRecurrence[{3, 9, 27, 81}, {3, 27, 189, 1215}, 50] (* G. C. Greubel, Dec 19 2017 *) -
PARI
x='x+O('x^30); Vec(-3*x*(108*x^3 +27*x^2 +6*x +1)/(81*x^4 +27*x^3 +9*x^2 +3*x -1)) \\ G. C. Greubel, Dec 19 2017
Formula
a(n) = Trace of matrix [({3,3,3,3},{3,0,0,0},{0,3,0,0},{0,0,3,0})^n].
a(n) = 3^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) = 3*a(n-1) + 9*a(n-2) + 27*a(n-3) + 81*a(n-4).
G.f.: -3*x*(108*x^3+27*x^2+6*x+1)/(81*x^4+27*x^3+9*x^2+3*x-1). (End)
Extensions
More terms from Colin Barker, Sep 02 2013