OFFSET
1,5
COMMENTS
A composition of n is a finite sequence of positive integers summing to n.
EXAMPLE
The distinct consecutive subsequences of (1,1,3,3) are (1), (1,1), (3), (1,3), (1,1,3), (3,3), (1,3,3), (1,1,3,3), all of which have different sums, so (1,1,3,3) is counted under a(8).
Triangle begins:
1
1 1
1 2 1
1 3 0 1
1 4 4 0 1
1 5 5 0 0 1
1 6 12 4 0 0 1
1 7 12 5 0 0 0 1
1 8 25 8 4 0 0 0 1
1 9 24 12 3 0 0 0 0 1
1 10 40 32 8 4 0 0 0 0 1
1 11 41 41 6 3 0 0 0 0 0 1
1 12 60 76 14 4 4 0 0 0 0 0 1
1 13 60 88 16 6 3 0 0 0 0 0 0 1
Row n = 8 counts the following compositions:
(8) (17) (116) (1115) (11111111)
(26) (125) (1133)
(35) (143) (2222)
(44) (152) (3311)
(53) (215) (5111)
(62) (233)
(71) (251)
(332)
(341)
(512)
(521)
(611)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n, {k}], UnsameQ@@Total/@Union[ReplaceList[#, {___, s__, ___}:>{s}]]&]], {n, 15}, {k, n}]
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, May 13 2019
STATUS
approved