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

%I A144677 #29 Sep 08 2022 08:45:38
%S A144677 1,2,3,6,9,12,18,24,30,40,50,60,75,90,105,126,147,168,196,224,252,288,
%T A144677 324,360,405,450,495,550,605,660,726,792,858,936,1014,1092,1183,1274,
%U A144677 1365,1470,1575,1680,1800,1920,2040,2176,2312,2448,2601,2754,2907,3078,3249,3420
%N A144677 Related to enumeration of quantum states (see reference for precise definition).
%C A144677 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
%C A144677 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
%H A144677 G. C. Greubel, <a href="/A144677/b144677.txt">Table of n, a(n) for n = 0..1000</a>
%H A144677 Brian O'Sullivan and Thomas Busch, <a href="http://arxiv.org/abs/0810.0231">Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas</a>, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10b, lambda=3]
%H A144677 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,2,-4,2,-1,2,-1).
%F A144677 From _Johannes W. Meijer_, May 20 2011: (Start)
%F A144677 a(n) = A190717(n-2) + A190717(n-1) + A190717(n).
%F A144677 a(n-2) + a(n-1) + a(n) = A014125(n).
%F A144677 G.f.: 1/((1-x)^4*(1+x+x^2)^2). (End)
%F A144677 From _Wesley Ivan Hurt_, Mar 28 2015: (Start)
%F A144677 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).
%F A144677 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)
%F A144677 a(n) = Sum_{j=0..n} floor((j+3)/3)*floor((n-j+3)/3). - _G. C. Greubel_, Oct 18 2021
%p A144677 n:=80; lambda:=3; S10b:=[];
%p A144677 for ii from 0 to n do
%p A144677 x:=floor(ii/lambda);
%p A144677 snc:=1/6*(x+1)*(x+2)*(3*ii-2*x*lambda+3);
%p A144677 S10b:=[op(S10b),snc];
%p A144677 od:
%p A144677 S10b;
%p A144677 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
%t A144677 CoefficientList[Series[1/((x - 1)^4*(x^2 + x + 1)^2), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Mar 28 2015 *)
%t A144677 LinearRecurrence[{2, -1, 2, -4, 2, -1, 2, -1}, {1, 2, 3, 6, 9, 12, 18, 24}, 60 ] (* _Vincenzo Librandi_, Mar 28 2015 *)
%o A144677 (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
%o A144677 (Sage)
%o A144677 @CachedFunction
%o A144677 def a(n): return sum( ((j+3)//3)*((n-j+3)//3) for j in (0..n) )
%o A144677 [a(n) for n in (0..60)] # _G. C. Greubel_, Oct 18 2021
%Y A144677 Cf. A006918, A144678, A144679.
%Y A144677 Cf. A000292, A190717. [_Johannes W. Meijer_, May 20 2011]
%K A144677 nonn,easy
%O A144677 0,2
%A A144677 _N. J. A. Sloane_, Feb 06 2009