OFFSET
0,3
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 partition y = (3,2,1,1) has semi-sums {2,3,4,5}, which is an interval, so y is counted under a(7).
The a(1) = 1 through a(8) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(211) (221) (51) (61) (62)
(1111) (2111) (222) (322) (71)
(11111) (321) (2221) (332)
(2211) (3211) (2222)
(21111) (22111) (3221)
(111111) (211111) (22211)
(1111111) (32111)
(221111)
(2111111)
(11111111)
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