A167993 Expansion of x^2/((3*x-1)*(3*x^2-1)).

0, 0, 1, 3, 12, 36, 117, 351, 1080, 3240, 9801, 29403, 88452, 265356, 796797, 2390391, 7173360, 21520080, 64566801, 193700403, 581120892, 1743362676, 5230147077, 15690441231, 47071500840, 141214502520, 423644039001, 1270932117003, 3812797945332, 11438393835996

Offset: 0

Author

Paul Curtz, Nov 16 2009

Keywords

• nonn
• easy

Comments

The terms satisfy a(n) = 3*a(n-1) +3*a(n-2) -9*a(n-3), so they follow the pattern a(n) = p*a(n-1) +q*a(n-2) -p*q*a(n-3) with p=q=3. This could be called the principal sequence for that recurrence because we have set all but one of the initial terms to zero. [p=q=1 leads to the principal sequence A004526. p=q=2 leads essentially to A032085. The common feature is that the denominator of the generating function does not have a root at x=1, so the sequences of higher order successive differences have the same recurrence as the original sequence. See A135094, A010036, A006516.]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (3,3,-9).

Crossrefs

Cf. A138587, A107767 (partial sums).

Programs

Mathematica

CoefficientList[Series[x^2/((3*x - 1)*(3*x^2 - 1)), {x, 0, 50}], x] (* G. C. Greubel, Jul 03 2016 *)
LinearRecurrence[{3,3,-9},{0,0,1},30] (* Harvey P. Dale, Nov 05 2017 *)

PARI

Vec(x^2/((3*x-1)*(3*x^2-1))+O(x^99)) \\ Charles R Greathouse IV, Jun 29 2011

Formula

a(2*n+1) = 3*a(2*n).

a(2*n) = A122006(2*n)/2.

a(n) = 3*a(n-1) + 3*a(n-4) - 9*a(n-3).

a(n+1) - a(n) = A122006(n).

a(n) = (3^n - A108411(n+1))/6.

G.f.: x^2/((3*x-1)*(3*x^2-1)).

From Colin Barker, Sep 23 2016: (Start)

a(n) = 3^(n-1)/2-3^(n/2-1)/2 for n even.

a(n) = 3^(n-1)/2-3^(n/2-1/2)/2 for n odd.

(End)

Extensions

Formulae corrected by Johannes W. Meijer, Jun 28 2011