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Number of subsets of {1..n} containing n but not containing the sum of any two distinct elements.
15

%I #10 Jan 13 2024 16:46:03

%S 0,1,2,3,6,9,15,24,41,60,99,149,236,355,552,817,1275,1870,2788,4167,

%T 6243,9098,13433,19718,28771,42137,60652,88603,127555,185200,261781,

%U 382931,541022,783862,1096608,1595829,2217467,3223064,4441073,6465800,8893694

%N Number of subsets of {1..n} containing n but not containing the sum of any two distinct elements.

%H Andrew Howroyd, <a href="/A364755/b364755.txt">Table of n, a(n) for n = 0..75</a>

%F First differences of A085489.

%e The subset S = {1,3,6,8} has pair-sums {4,7,9,11,14}, which are disjoint from S, so it is counted under a(8).

%e The a(1) = 1 through a(6) = 15 subsets:

%e {1} {2} {3} {4} {5} {6}

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

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

%e {3,4} {3,5} {3,6}

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

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

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

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

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

%e {2,3,6}

%e {2,5,6}

%e {3,4,6}

%e {3,5,6}

%e {4,5,6}

%e {3,4,5,6}

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

%Y Partial sums are A085489(n) - 1, complement counted by A364534.

%Y With re-usable parts we have A288728.

%Y The complement with n is counted by A364756, first differences of A088809.

%Y Cf. A007865, A050291, A054519, A093971, A151897, A236912, A326020, A326080, A326083, A364272, A364349, A364533.

%K nonn

%O 0,3

%A _Gus Wiseman_, Aug 11 2023

%E a(21) onwards added (using A085489) by _Andrew Howroyd_, Jan 13 2024