A122008 Expansion of (2*x-1)*(x-1)*x / ((3*x-1)*(3*x^2-1)).

1, 0, 5, 6, 33, 72, 261, 702, 2241, 6480, 19845, 58806, 177633, 530712, 1595781, 4780782, 14353281, 43040160, 129153285, 387400806, 1162300833, 3486725352, 10460471301, 31380882462, 94143533121, 282429005040, 847289672325, 2541864234006, 7625600673633

Offset

1,3

Comments

Limit_{n->infinity} a(n+1)/a(n) = 3.

References

Alain M. Robert, Linear Algebra, Examples and Applications World Scientific, 2005, p. 58.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,3,-9).

Formula

a(n) = 3^(n-2) - (-1)^n*A038754(n)/3. - R. J. Mathar, Nov 07 2011

From Colin Barker, Feb 05 2017: (Start)

a(n) = 3^(n-2) - 3^((n-3)/2)/2*(-2 + 2*(-1)^n + sqrt(3) + (-1)^n*sqrt(3)) for n>0.

a(n) = 3*a(n-1) + 3*a(n-2) - 9*a(n-3) for n>3.

(End)

Maple

A038754 := proc(n)
        if type(n, 'even') then
                3^(n/2);
        else
                2*3^((n-1)/2) ;
        end if;
end proc:
A122008 := proc(n)
        3^(n-2)-(-1)^n*A038754(n)/3 ;
end proc:
seq(A122008(n), n=1..10) ; # R. J. Mathar, Nov 07 2011

Mathematica

M = {{0, 1, 2}, {1, 2, 0}, {2, 0, 1}} v[1] = {1, 0, 0} v[n_] := v[n] = M.v[n - 1] a1 = Table[v[n][[1]], {n, 1, 50}]
LinearRecurrence[{3, 3, -9}, {1, 0, 5}, 30] (* Harvey P. Dale, Aug 28 2023 *)

Prog

(PARI) Vec((2*x-1)*(x-1)*x / ((3*x-1)*(3*x^2-1)) + O(x^50)) \\ Colin Barker, Feb 05 2017

Crossrefs

Sequence in context: A137762 A177118 A151506 * A248254 A212918 A255197

Adjacent sequences: A122005 A122006 A122007 * A122009 A122010 A122011

Keyword

nonn,easy

Author

Roger L. Bagula and Gary W. Adamson, Sep 11 2006

Status

approved