A365071 - OEIS

A365071

Number of subsets of {1..n} containing n such that no element is a sum of distinct other elements. A variation of non-binary sum-free subsets without re-usable elements.

0, 1, 2, 3, 6, 9, 15, 23, 40, 55, 94, 132, 210, 298, 476, 644, 1038, 1406, 2149, 2965, 4584, 6077, 9426, 12648, 19067, 25739, 38958, 51514, 78459, 104265, 155436, 208329, 312791, 411886, 620780, 823785, 1224414, 1631815, 2437015, 3217077, 4822991

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OFFSET

0,3

COMMENTS

The complement is counted by A365069. The binary version is A364755, complement A364756. For re-usable parts we have A288728, complement A365070.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..85

S. R. Finch, Monoids of natural numbers, March 17, 2009.

FORMULA

a(n) + A365069(n) = 2^(n-1).

First differences of A151897.

EXAMPLE

The subset {1,3,4,6} has 4 = 1 + 3 so is not counted under a(6).

The subset {2,3,4,5,6} has 6 = 2 + 4 and 4 = 1 + 3 so is not counted under a(6).

The a(0) = 0 through a(6) = 15 subsets:

. {1} {2} {3} {4} {5} {6}

{1,2} {1,3} {1,4} {1,5} {1,6}

{2,3} {2,4} {2,5} {2,6}

{3,4} {3,5} {3,6}

{1,2,4} {4,5} {4,6}

{2,3,4} {1,2,5} {5,6}

{1,3,5} {1,2,6}

{2,4,5} {1,3,6}

{3,4,5} {1,4,6}

{2,3,6}

{2,5,6}

{3,4,6}

{3,5,6}

{4,5,6}

{3,4,5,6}

MATHEMATICA

Table[Length[Select[Subsets[Range[n]], MemberQ[#, n]&&Intersection[#, Total/@Subsets[#, {2, Length[#]}]]=={}&]], {n, 0, 10}]

CROSSREFS

First differences of A151897.

The version with re-usable parts is A288728 first differences of A007865.

The binary version is A364755, first differences of A085489.

The binary complement is A364756, first differences of A088809.

The complement is counted by A365069, first differences of A364534.

The complement w/ re-usable parts is A365070, first differences of A093971.

A108917 counts knapsack partitions, strict A275972.

A124506 counts combination-free subsets, differences of A326083.

A364350 counts combination-free strict partitions, complement A364839.

A365046 counts combination-full subsets, differences of A364914.

Partitions: A236912, A237113, A237668, A364532, A364272, A364349, A364913.

Cf. A050291, A095944, A103580, A324741, A326080, A326117, A341507, A364533.

Sequence in context: A217067 A014214 A094993 * A192671 A080239 A364755

Adjacent sequences: A365068 A365069 A365070 * A365072 A365073 A365074

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 26 2023

EXTENSIONS

a(14) onwards added (using A151897) by Andrew Howroyd, Jan 13 2024

STATUS

approved