OFFSET
0,3
COMMENTS
From Enrique Navarrete, Jul 09 2024: (Start)
a(n+1) is the number of generalized compositions of n using parts of size at most 3 where there are binomial(3,i) types of i.
For example, the following table gives the type of composition, the number of such compositions, and the total number of compositions of n = 5 using parts of size at most 3 where there are binomial(3,i) types of i (ie. 3 types of 1, 3 types of 2 and 1 type of 3):
Type Number Total
3+2 2 6
3+1+1 3 27
2+2+1 3 81
2+1+1+1 4 324
1+1+1+1+1 1 243,
adding to a(6) = 681.
The coefficients of 1/(1 - C(k,1)*x - C(k,2)*x^2 - C(k,3)*x^3 - ... - C(k,k)*x^k) give the number of generalized compositions of n using parts of size at most k where there are binomial(k,i) types of i. (End).
For n>0, the expansion of (4^(1/3) + 2^(1/3) + 1)^n is a(n)*4^(1/3) + (a(n) + a(n-1))*2^(1/3) + (a(n) + 2*a(n-1) + a(n-2)). - Greg Dresden, Aug 14 2024
REFERENCES
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 562.
LINKS
FORMULA
x=a(n), z=a(-n), y=a(n)+a(n-1), t=a(-n)+a(-n-1) is a solution to 2(x^3+z^3)=y^3+t^3.
G.f.: x/(1-3*x-3*x^2-x^3).
a(n) = 3*a(n-1)+3*a(n-2)+a(n-3).
a(-1-n) = A108369(n).
a(n+1) = Sum_{k>=0} (1/2)^(k+1) * binomial(3*k,n). - Seiichi Manyama, Aug 03 2024
MATHEMATICA
CoefficientList[Series[x/(1-3*x-3*x^2-x^3), {x, 0, 40}], x] (* or *) LinearRecurrence[{3, 3, 1}, {0, 1, 3}, 40] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)
PROG
(PARI) a(n)=if(n>=0, polcoeff(x/(1-3*x-3*x^2-x^3)+x*O(x^n), n), n=-1-n; polcoeff(x/(1+3*x+3*x^2-x^3)+x*O(x^n), n))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Jun 01 2005
STATUS
approved