OFFSET
0,7
COMMENTS
We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.
EXAMPLE
The a(0) = 0 through a(9) = 13 partitions:
. . . . . (311) (411) (331) (422) (441)
(3111) (421) (431) (522)
(511) (521) (531)
(4111) (611) (621)
(31111) (3311) (711)
(4211) (4311)
(5111) (5211)
(41111) (6111)
(311111) (33111)
(42111)
(51111)
(411111)
(3111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], (d=Total/@Subsets[#, {2}]; If[d=={}, {}, Range[Min@@d, Max@@d]]!=Union[d])&]], {n, 0, 15}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 17 2023
STATUS
approved