|
|
A008679
|
|
Expansion of 1/((1-x^3)*(1-x^4)).
|
|
6
|
|
|
1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 6, 5, 5, 6, 6, 5, 6, 6, 6, 6, 6, 6, 7, 6, 6, 7, 7, 6, 7, 7, 7, 7, 7, 7, 8, 7
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,13
|
|
COMMENTS
|
With four 0's prepended and offset 0, a(n) is the number of partitions of n into four parts whose largest three parts are equal. - Wesley Ivan Hurt, Jan 06 2021
|
|
LINKS
|
|
|
FORMULA
|
G.f.: 1/((1-x)^2*(1+x)*(1+x+x^2)*(1+x^2)). - R. J. Mathar, Feb 13 2009
a(n) = 1 + floor(n/3) + floor(-n/4). - Tani Akinari, Sep 02 2013
E.g.f.: (1/72)*(9*exp(-x)+21*exp(x)+6*exp(x)*x+18*cos(x)+24*exp(-x/2)*cos(sqrt(3)*x/2)-18*sin(x)+8*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2)). - Stefano Spezia, Sep 09 2019
a(n) = a(n-3) + a(n-4) - a(n-7).
a(n) = Sum_{k=1..floor((n+4)/4)} Sum_{j=k..floor((n+4-k)/3)} Sum_{i=j..floor((n+4-j-k)/2)} [j = i = n+4-i-k-j], where [ ] is the Iverson bracket. (End)
|
|
MAPLE
|
seq(coeff(series(1/((1-x^3)*(1-x^4)), x, n+1), x, n), n = 0..90); # G. C. Greubel, Sep 09 2019
|
|
MATHEMATICA
|
LinearRecurrence[{0, 0, 1, 1, 0, 0, -1}, {1, 0, 0, 1, 1, 0, 1}, 90] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
CoefficientList[Series[1/((1-x)^2(1+x)(1+x+x^2)(1+x^2)), {x, 0, 90}], x] (* Vincenzo Librandi, Jun 11 2013 *)
|
|
PROG
|
(PARI) my(x='x+O('x^90)); Vec(1/((1-x^3)*(1-x^4))) \\ G. C. Greubel, Sep 09 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 90); Coefficients(R!( 1/((1-x^3)*(1-x^4)) )); // G. C. Greubel, Sep 09 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P(1/((1-x^3)*(1-x^4))).list()
(GAP) a:=[1, 0, 0, 1, 1, 0, 1, 1];; for n in [8..90] do a[n]:=a[n-3]+a[n-4]-a[n-7]; od; a; # G. C. Greubel, Sep 09 2019
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|