login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A144678 Related to enumeration of quantum states (see reference for precise definition). 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
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]
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)
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 *)
PROG
(PARI) Vec(1/(x-1)^4/(x^3+x^2+x+1)^2+O(x^99)) \\ Charles R Greathouse IV, Jun 20 2013
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1-x^4))^2 )); // G. C. Greubel, Oct 18 2021
(Sage)
def A144678_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 1/((1-x)*(1-x^4))^2 ).list()
A144678_list(60) # G. C. Greubel, Oct 18 2021
CROSSREFS
Sequence in context: A062042 A107817 A008811 * A309678 A279225 A073149
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Feb 06 2009
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 21 12:41 EDT 2024. Contains 373544 sequences. (Running on oeis4.)