A239288 Maximal sum of x0 + x0*x1 + ... + x0*x1*...*xk over all compositions x0 + ... + xk = n.

0, 1, 2, 4, 6, 10, 15, 22, 33, 48, 69, 102, 147, 210, 309, 444, 633, 930, 1335, 1902, 2793, 4008, 5709, 8382, 12027, 17130, 25149, 36084, 51393, 75450, 108255, 154182, 226353, 324768, 462549, 679062, 974307, 1387650, 2037189, 2922924, 4162953, 6111570

Offset: 0

Author

Thomas Dybdahl Ahle, Jun 13 2014

Keywords

• nonn
• easy

Comments

This sequence comes up in the analysis of exact algorithms for maximum independent sets.

Examples

			For n = 4 there are three solutions, all summing to 6: 3+3*1, 2+2*1+2*1*1, 2+2*2.
For n = 7 there is only one solution: 2 + 2*2 + 2*2*2 + 2*2*2*1.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3).

Programs

Maple

mulprod := proc(L)
    local i,k ;
    add(mul(op(k,L),k=1..i),i=1..nops(L)) ;
end proc:
A239288 := proc(n)
    a := 0 ;
    for pa in combinat[partition](n) do
        for pe in combinat[permute](pa) do
            f := mulprod(pe) ;
            a := max(a,f) ;
        end do:
    end do:
    return a;
end proc: # R. J. Mathar, Jul 03 2014

Formula

a(0) = 0, a(n) = max{k + k*a(n - k) | 1 <= k <= n}.

For n >= 8 the solution becomes cyclic: a(3n + k) = 3 + 3a(3n - 3 + k).

G.f.: -x*(x^2+x+1)*(x^5-2*x^4+2*x^3-x^2-1) / ((x-1)*(3*x^3-1)). - Joerg Arndt