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

%I #29 Sep 08 2022 08:44:36

%S 1,0,2,2,3,4,6,6,9,10,12,14,17,18,22,24,27,30,34,36,41,44,48,52,57,60,

%T 66,70,75,80,86,90,97,102,108,114,121,126,134,140,147,154,162,168,177,

%U 184,192,200,209,216,226,234,243,252,262,270,281,290,300,310,321

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

%H G. C. Greubel, <a href="/A008806/b008806.txt">Table of n, a(n) for n = 0..1000</a>

%H David Beckwith, Vadim Ponomarenko and Rob Pratt, <a href="http://www.jstor.org/stable/27642265">Building Two Piles of Equal Height: 11183</a>, The American Mathematical Monthly, 114 (2007), 551-552.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).

%F From _R. J. Mathar_, Nov 08 2010: (Start)

%F a(n) = (16*A131713(n) +29 +24*n +6*n^2 +27*(-1)^n)/72.

%F G.f.: (1 -x +x^2)/( (1+x)*(1+x+x^2)*(1-x)^3 ). (End)

%F a(n) = floor((6*n^2+24*n+61+27*(-1)^n)/72). - _Tani Akinari_, Jul 24 2013

%p seq(coeff(series((1+x^3)/((1-x^2)^2*(1-x^3)), x, n+1), x, n), n = 0..70); # _G. C. Greubel_, Sep 12 2019

%t CoefficientList[Series[(1+x^3)/((1-x^2)^2*(1-x^3)), {x,0,70}], x] (* or *) LinearRecurrence[{1,1,0,-1,-1,1}, {1,0,2,2,3,4}, 70] (* _G. C. Greubel_, Sep 12 2019 *)

%o (PARI) Vec((1+x^3)/((1-x^2)^2*(1-x^3)) +O(x^70)) \\ _Charles R Greathouse IV_, Sep 26 2012; modified by _G. C. Greubel_, Sep 12 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^3)/((1-x^2)^2*(1-x^3)) )); // _G. C. Greubel_, Sep 12 2019

%o (Sage)

%o def A008806_list(prec):

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

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

%o A008806_list(70) # _G. C. Greubel_, Sep 12 2019

%o (GAP) a:=[1,0,2,2,3,4];; for n in [7..70] do a[n]:=a[n-1]+a[n-2]-a[n-4]-a[n-5]+a[n-6]; od; a; # _G. C. Greubel_, Sep 12 2019

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

%E Terms a(52) onward added by _G. C. Greubel_, Sep 12 2019