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

1, 2, 3, 4, 7, 10, 13, 16, 22, 28, 34, 40, 50, 60, 70, 80, 95, 110, 125, 140, 161, 182, 203, 224, 252, 280, 308, 336, 372, 408, 444, 480, 525, 570, 615, 660, 715, 770, 825, 880, 946, 1012, 1078, 1144, 1222, 1300, 1378, 1456, 1547, 1638, 1729, 1820, 1925, 2030, 2135

Offset: 0

Author

N. J. A. Sloane, Feb 06 2009

Keywords

• nonn
• easy

Comments

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]

Partial sums of 1,1,1,1, 3,3,3,3, 6,6,6,6,..., the quadruplicated A000217. - R. J. Mathar, Aug 25 2013

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

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=4]
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,2,-4,2,0,-1,2,-1).

Crossrefs

Cf. A000292, A006918, A144677, A144679, A190718.

Programs

Magma

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1-x^4))^2 )); // G. C. Greubel, Oct 18 2021

Maple

n:=80; lambda:=4; 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;
A144678 := proc(n) option remember;
   local k;
   sum(A190718(n-k),k=0..3)
end:
A190718:= proc(n)
   binomial(floor(n/4)+3,3)
end:
seq(A144678(n),n=0..54); # Johannes W. Meijer, May 20 2011

Mathematica

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 *)
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 *)

PARI

Vec(1/(x-1)^4/(x^3+x^2+x+1)^2+O(x^99)) \\ Charles R Greathouse IV, Jun 20 2013

Sage

def A144678_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)*(1-x^4))^2 ).list()
A144678_list(60) # G. C. Greubel, Oct 18 2021

Formula

From Johannes W. Meijer, May 20 2011: (Start)

a(n) = A190718(n-3) + A190718(n-2) + A190718(n-1) + A190718(n).

a(n-3) + a(n-2) + a(n-1) + a(n) = A122046(n+3).

G.f.: 1/((x-1)^4*(x^3+x^2+x+1)^2). (End)

a(n) = A009531(n+5)/16 + (n+5)*(2*n^2+20*n+33+3*(-1)^n)/192 . - R. J. Mathar, Jun 20 2013

a(n) = Sum_{i=1..n+8} floor(i/4) * floor((n+8-i)/4). - Wesley Ivan Hurt, Jul 21 2014

From Alois P. Heinz, Dec 22 2021: (Start)

G.f.: 1/((1-x)*(1-x^4))^2.

a(n) = Sum_{j=0..floor(n/4)} (j+1)*(n-4*j+1). (End)