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Expansion of 1/((1-x)^3*(1-x^3)^2).
4

%I #32 Oct 23 2024 00:45:47

%S 1,3,6,12,21,33,51,75,105,145,195,255,330,420,525,651,798,966,1162,

%T 1386,1638,1926,2250,2610,3015,3465,3960,4510,5115,5775,6501,7293,

%U 8151,9087,10101,11193,12376,13650

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

%C 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

%H Vincenzo Librandi, <a href="/A011779/b011779.txt">Table of n, a(n) for n = 0..1000</a>

%H Project Euler, <a href="https://projecteuler.net/problem=577">Problem 577. Counting hexagons</a>.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,3,-6,6,-3,3,-3,1).

%F 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

%t CoefficientList[Series[1 / ((1 - x)^3 (1 - x^3)^2), {x, 0, 100}], x] (* _Vincenzo Librandi_, Jun 23 2013 *)

%o (PARI) Vec(1/((1-x)^3*(1-x^3)^2)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 25 2012

%o (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

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 60);

%o Coefficients(R!( 1/((1-x)^3*(1-x^3)^2) )); // _G. C. Greubel_, Oct 22 2024

%o (SageMath)

%o def A011779_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( 1/((1-x)^3*(1-x^3)^2) ).list()

%o A011779_list(60) # _G. C. Greubel_, Oct 22 2024

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

%K nonn,easy,changed

%O 0,2

%A _Emeric Deutsch_