A011779 Expansion of 1/((1-x)^3*(1-x^3)^2).

1, 3, 6, 12, 21, 33, 51, 75, 105, 145, 195, 255, 330, 420, 525, 651, 798, 966, 1162, 1386, 1638, 1926, 2250, 2610, 3015, 3465, 3960, 4510, 5115, 5775, 6501, 7293, 8151, 9087, 10101, 11193, 12376, 13650

Offset: 0

Author

Emeric Deutsch

Keywords

• nonn
• easy

Comments

The Ca2 and Ze4 triangle sums of A139600 are related to the sequence given above, e.g., Ze4(n) = A011779(n-1) - A011779(n-2) - A011779(n-4) + 3*A011779(n-5), with A011779(n) = 0 for n <= -1. For the definitions of these triangle sums see A180662. - Johannes W. Meijer, Apr 29 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Project Euler, Problem 577. Counting hexagons.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-6,6,-3,3,-3,1).

Crossrefs

Cf. A011779, A049347, A099254, A139600, A236770 (first trisection, except 0).

Programs

Magma

R:=PowerSeriesRing(Integers(), 60);
Coefficients(R!( 1/((1-x)^3*(1-x^3)^2) )); // G. C. Greubel, Oct 22 2024

Mathematica

CoefficientList[Series[1 / ((1 - x)^3 (1 - x^3)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Jun 23 2013 *)

PARI

Vec(1/((1-x)^3*(1-x^3)^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

PARI

a(n)=1/216 * n^4 + 1/12 * n^3 + 37/72 * n^2 + [5/4, 139/108, 131/108][1+n%3] * n + [1, 10/9, 7/9][1+n%3] \\ Yurii Ivanov, Jul 06 2021

SageMath

def A011779_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)^3*(1-x^3)^2) ).list()
A011779_list(60) # G. C. Greubel, Oct 22 2024

Formula

a(n) = (1/216)*((208 + 270*n + 111*n^2 + 18*n^3 + n^4) - 8*(-1)^n*(A099254(n) + A099254(n-1)) + 16*(A049347(n) + 2*A049347(n-1)) ). - G. C. Greubel, Oct 22 2024