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A365042
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Number of subsets of {1..n} containing n such that some element can be written as a positive linear combination of the others.
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8
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0, 0, 1, 2, 4, 5, 9, 11, 17, 21, 29, 36, 50, 60, 78, 95, 123, 147, 185, 221, 274, 325, 399, 472, 574, 672, 810, 945, 1131, 1316, 1557, 1812, 2137, 2462, 2892, 3322, 3881, 4460, 5176, 5916, 6846, 7817, 8993, 10250, 11765, 13333, 15280, 17308, 19731, 22306
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OFFSET
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0,4
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COMMENTS
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Sets of this type may be called "positive combination-full".
Also subsets of {1..n} containing n whose greatest element can be written as a positive linear combination of the others.
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LINKS
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FORMULA
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EXAMPLE
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The subset {3,4,10} has 10 = 2*3 + 1*4 so is counted under a(10).
The a(0) = 0 through a(7) = 11 subsets:
. . {1,2} {1,3} {1,4} {1,5} {1,6} {1,7}
{1,2,3} {2,4} {1,2,5} {2,6} {1,2,7}
{1,2,4} {1,3,5} {3,6} {1,3,7}
{1,3,4} {1,4,5} {1,2,6} {1,4,7}
{2,3,5} {1,3,6} {1,5,7}
{1,4,6} {1,6,7}
{1,5,6} {2,3,7}
{2,4,6} {2,5,7}
{1,2,3,6} {3,4,7}
{1,2,3,7}
{1,2,4,7}
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MATHEMATICA
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combp[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 1, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n]], MemberQ[#, n]&&Or@@Table[combp[#[[k]], Union[Delete[#, k]]]!={}, {k, Length[#]}]&]], {n, 0, 10}]
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CROSSREFS
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Without re-usable parts we have A365069, first differences of A364534.
A085489 and A364755 count subsets with no sum of two distinct elements.
A088314 counts sets that can be linearly combined to obtain n.
A088809 and A364756 count subsets with some sum of two distinct elements.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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