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%I #14 Jan 13 2024 16:47:15
%S 0,0,1,1,5,9,24,46,109,209,469,922,1932,3858,7952,15831,32214,64351,
%T 129813,259566,521681,1042703,2091626,4182470,8376007,16752524,
%U 33530042,67055129,134165194,268328011,536763582,1073523097,2147268041,4294505929,8589506814,17178978145
%N Number of subsets of {1..n} containing n and some element equal to the sum of two other (possibly equal) elements.
%C These are binary sum-full sets where elements can be re-used. The complement is counted by A288728. The non-binary version is A365046, complement A124506. For non-re-usable parts we have A364756, complement A085489.
%H Andrew Howroyd, <a href="/A365070/b365070.txt">Table of n, a(n) for n = 0..88</a>
%H S. R. Finch, <a href="/A066062/a066062.pdf">Monoids of natural numbers</a>, March 17, 2009.
%F First differences of A093971.
%e The subset {1,3} has no element equal to the sum of two others, so is not counted under a(3).
%e The subset {3,4,5} has no element equal to the sum of two others, so is not counted under a(5).
%e The subset {1,3,4} has 4 = 1 + 3, so is counted under a(4).
%e The subset {2,4,5} has 4 = 2 + 2, so is counted under a(5).
%e The a(0) = 0 through a(5) = 9 subsets:
%e . . {1,2} {1,2,3} {2,4} {1,2,5}
%e {1,2,4} {1,4,5}
%e {1,3,4} {2,3,5}
%e {2,3,4} {2,4,5}
%e {1,2,3,4} {1,2,3,5}
%e {1,2,4,5}
%e {1,3,4,5}
%e {2,3,4,5}
%e {1,2,3,4,5}
%t Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&Intersection[#,Total /@ Tuples[#,2]]!={}&]], {n,0,10}]
%Y The complement w/o re-usable parts is A085489, first differences of A364755.
%Y First differences of A093971.
%Y The non-binary complement is A124506, first differences of A326083.
%Y The complement is counted by A288728, first differences of A007865.
%Y For partitions (not requiring n) we have A363225, strict A363226.
%Y The case without re-usable parts is A364756, firsts differences of A088809.
%Y The non-binary version is A365046, first differences of A364914.
%Y A116861 and A364916 count linear combinations of strict partitions.
%Y A364350 counts combination-free strict partitions, complement A364839.
%Y A364913 counts combination-full partitions.
%Y A365006 counts no positive combination-full strict ptns.
%Y Cf. A050291, A051026, A151897, A236912, A237113, A237668, A326080, A364349, A364533, A364670.
%K nonn
%O 0,5
%A _Gus Wiseman_, Aug 24 2023
%E a(21) onwards added (using A093971) by _Andrew Howroyd_, Jan 13 2024
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