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A144677 Related to enumeration of quantum states (see reference for precise definition). 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Equals (1, 2, 3, ...) convolved with (1, 0, 0, 2, 0, 0, 3, ...) = (1 + 2x + 3x^2 + ...) * (1 + 2x^3 + 3x^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

Table of n, a(n) for n=0..53.

Brian OSullivan 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/((x-1)^4*(x^2+x+1)^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)

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

CROSSREFS

Cf. A006918, A144678, A144679.

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

Sequence in context: A280984 A008810 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

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Last modified December 6 21:48 EST 2019. Contains 329809 sequences. (Running on oeis4.)