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A239288
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Maximal sum of x0 + x0*x1 + ... + x0*x1*...*xk over all compositions x0 + ... + xk = n.
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1
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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
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OFFSET
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0,3
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COMMENTS
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This sequence comes up in the analysis of exact algorithms for maximum independent sets.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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mulprod := proc(L)
local i, k ;
add(mul(op(k, L), k=1..i), i=1..nops(L)) ;
end proc:
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;
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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