%I #16 May 24 2024 15:28:08

%S 1,1,4,5,8,11,15,18,24,28,34,40,47,53,62,69,78,87,97,106,118,128,140,

%T 152,165,177,192,205,220,235,251,266,284,300,318,336,355,373,394,413,

%U 434,455,477,498,522,544,568,592,617,641,668

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

%C Partial sums of A117572.

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

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

%F a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6).

%F a(n) = sqrt(3)*cos(2*Pi*n/3+Pi/6)/9 - sin(2*Pi*n/3+Pi/6)/3 + 3*cos(Pi*n)/8 + (6n^2+20n+15)/24.

%F a(n) = floor((9*(-1)^n + 6*n^2 + 20*n + 23)/24). - _Tani Akinari_, Nov 09 2012

%F a(n) = 1 - n/4 + n^2/4 + 3/2*floor(n/2) + floor((n+1)/3). - _Vaclav Kotesovec_, Jun 15 2014

%t CoefficientList[Series[(1+2x^2)/((1-x)(1-x^2)(1-x^3)),{x,0,50}],x] (* or *) LinearRecurrence[{1,1,0,-1,-1,1},{1,1,4,5,8,11},60] (* _Harvey P. Dale_, Jun 06 2013 *)

%Y Cf. A117572.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Mar 29 2006