OFFSET
0,2
COMMENTS
Equals (1, 2, 3, ...) convolved with (1, 0, 0, 2, 0, 0, 3, ...) = (1 + 2*x + 3*x^2 + ...) * (1 + 2*x^3 + 3*x^6 + ...). - Gary W. Adamson, Feb 23 2010
The Ca2 and Ze4 triangle sums, see A180662 for their definitions, of the Connell-Pol triangle A159797 are linear sums of shifted versions of the sequence given above, e.g., Ca2(n) = a(n-1) + 2*a(n-2) + 3*a(n-3) + a(n-4). - Johannes W. Meijer, May 20 2011
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10b, lambda=3]
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).
FORMULA
From Johannes W. Meijer, May 20 2011: (Start)
a(n-2) + a(n-1) + a(n) = A014125(n).
G.f.: 1/((1-x)^4*(1+x+x^2)^2). (End)
From Wesley Ivan Hurt, Mar 28 2015: (Start)
a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -4*a(n-4) +2*a(n-5) -a(n-6) +2*a(n-7) -a(n-8).
a(n) = ((2 + floor(n/3))^3 - floor((n+4)/3) + floor((n+4)/3)^3 - floor((n+5)/3) + floor((n+5)/3)^3 - floor((n+6)/3))/6. (End)
a(n) = Sum_{j=0..n} floor((j+3)/3)*floor((n-j+3)/3). - G. C. Greubel, Oct 18 2021
MAPLE
n:=80; lambda:=3; S10b:=[];
for ii from 0 to n do
x:=floor(ii/lambda);
snc:=1/6*(x+1)*(x+2)*(3*ii-2*x*lambda+3);
S10b:=[op(S10b), snc];
od:
S10b;
MATHEMATICA
CoefficientList[Series[1/((x - 1)^4*(x^2 + x + 1)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Mar 28 2015 *)
LinearRecurrence[{2, -1, 2, -4, 2, -1, 2, -1}, {1, 2, 3, 6, 9, 12, 18, 24}, 60 ] (* Vincenzo Librandi, Mar 28 2015 *)
PROG
(Magma) I:=[1, 2, 3, 6, 9, 12, 18, 24]; [n le 8 select I[n] else 2*Self(n-1)-Self(n-2)+2*Self(n-3)-4*Self(n-4)+2*Self(n-5)-Self(n-6)+2*Self(n-7)-Self(n-8): n in [1..60]]; // Vincenzo Librandi, Mar 28 2015
(Sage)
@CachedFunction
def a(n): return sum( ((j+3)//3)*((n-j+3)//3) for j in (0..n) )
[a(n) for n in (0..60)] # G. C. Greubel, Oct 18 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Feb 06 2009
STATUS
approved