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A144678 Related to enumeration of quantum states (see reference for precise definition).

%I A144678 #49 Sep 08 2022 08:45:38
%S A144678 1,2,3,4,7,10,13,16,22,28,34,40,50,60,70,80,95,110,125,140,161,182,
%T A144678 203,224,252,280,308,336,372,408,444,480,525,570,615,660,715,770,825,
%U A144678 880,946,1012,1078,1144,1222,1300,1378,1456,1547,1638,1729,1820,1925,2030,2135
%N A144678 Related to enumeration of quantum states (see reference for precise definition).
%C A144678 The Gi2 triangle sums of the triangle A159797 are linear sums of shifted versions of the sequence given above, i.e., Gi2(n) = a(n-1) + 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5). For the definitions of the Gi2 and other triangle sums see A180662. [_Johannes W. Meijer_, May 20 2011]
%C A144678 Partial sums of 1,1,1,1, 3,3,3,3, 6,6,6,6,..., the quadruplicated A000217. - _R. J. Mathar_, Aug 25 2013
%C A144678 Number of partitions of n into two different parts of size 4 and two different parts of size 1. a(4) = 7: 4, 4', 1111, 1111', 111'1', 11'1'1', 1'1'1'1'. - _Alois P. Heinz_, Dec 22 2021
%H A144678 G. C. Greubel, <a href="/A144678/b144678.txt">Table of n, a(n) for n = 0..1000</a>
%H A144678 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=4]
%H A144678 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,2,-4,2,0,-1,2,-1).
%F A144678 From _Johannes W. Meijer_, May 20 2011: (Start)
%F A144678 a(n) = A190718(n-3) + A190718(n-2) + A190718(n-1) + A190718(n).
%F A144678 a(n-3) + a(n-2) + a(n-1) + a(n) = A122046(n+3).
%F A144678 G.f.: 1/((x-1)^4*(x^3+x^2+x+1)^2). (End)
%F A144678 a(n) = A009531(n+5)/16 + (n+5)*(2*n^2+20*n+33+3*(-1)^n)/192 . - _R. J. Mathar_, Jun 20 2013
%F A144678 a(n) = Sum_{i=1..n+8} floor(i/4) * floor((n+8-i)/4). - _Wesley Ivan Hurt_, Jul 21 2014
%F A144678 From _Alois P. Heinz_, Dec 22 2021: (Start)
%F A144678 G.f.: 1/((1-x)*(1-x^4))^2.
%F A144678 a(n) = Sum_{j=0..floor(n/4)} (j+1)*(n-4*j+1). (End)
%p A144678 n:=80; lambda:=4; S10b:=[];
%p A144678 for ii from 0 to n do
%p A144678 x:=floor(ii/lambda);
%p A144678 snc:=1/6*(x+1)*(x+2)*(3*ii-2*x*lambda+3);
%p A144678 S10b:=[op(S10b),snc];
%p A144678 od:
%p A144678 S10b;
%p A144678 A144678 := proc(n) option remember;
%p A144678    local k;
%p A144678    sum(A190718(n-k),k=0..3)
%p A144678 end:
%p A144678 A190718:= proc(n)
%p A144678    binomial(floor(n/4)+3,3)
%p A144678 end:
%p A144678 seq(A144678(n),n=0..54); # _Johannes W. Meijer_, May 20 2011
%t A144678 a[n_] = (r = Mod[n, 4]; (4+n-r)(8+n-r)(3+n+2r)/96); Table[a[n], {n, 0, 54}] (* _Jean-François Alcover_, Sep 02 2011 *)
%t A144678 LinearRecurrence[{2,-1,0,2,-4,2,0,-1,2,-1}, {1,2,3,4,7,10,13,16,22,28}, 60] (* _G. C. Greubel_, Oct 18 2021 *)
%o A144678 (PARI) Vec(1/(x-1)^4/(x^3+x^2+x+1)^2+O(x^99)) \\ _Charles R Greathouse IV_, Jun 20 2013
%o A144678 (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1-x^4))^2 )); // _G. C. Greubel_, Oct 18 2021
%o A144678 (Sage)
%o A144678 def A144678_list(prec):
%o A144678     P.<x> = PowerSeriesRing(ZZ, prec)
%o A144678     return P( 1/((1-x)*(1-x^4))^2 ).list()
%o A144678 A144678_list(60) # _G. C. Greubel_, Oct 18 2021
%Y A144678 Cf. A000292, A006918, A144677, A144679, A190718.
%K A144678 nonn,easy
%O A144678 0,2
%A A144678 _N. J. A. Sloane_, Feb 06 2009