A144677 Related to enumeration of quantum states (see reference for precise definition).

1, 2, 3, 6, 9, 12, 18, 24, 30, 40, 50, 60, 75, 90, 105, 126, 147, 168, 196, 224, 252, 288, 324, 360, 405, 450, 495, 550, 605, 660, 726, 792, 858, 936, 1014, 1092, 1183, 1274, 1365, 1470, 1575, 1680, 1800, 1920, 2040, 2176, 2312, 2448, 2601, 2754, 2907, 3078, 3249, 3420

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) = A190717(n-2) + A190717(n-1) + A190717(n).

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;
A144677 := proc(n) option remember; local k1; sum(A190717(n-k1), k1=0..2) end: A190717:= proc(n) option remember; A190717(n):= binomial(floor(n/3)+3, 3) end: seq(A144677(n), n=0..53); # Johannes W. Meijer, May 20 2011

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

Cf. A006918, A144678, A144679.

Cf. A000292, A190717. [Johannes W. Meijer, May 20 2011]

Sequence in context: A280984 A339485 A176893 * A309677 A058616 A298435

Adjacent sequences: A144674 A144675 A144676 * A144678 A144679 A144680

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 06 2009

Status

approved