OFFSET
0,4
COMMENTS
A composition of n is a finite sequence of positive integers summing to n.
Compare to the definition of perfect partitions (A002033).
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100
Fausto A. C. Cariboni, All compositions that yield a(n) for n = 1..100, Feb 21 2022.
EXAMPLE
The distinct consecutive subsequences of (3,4,1,1) together with their sums are:
1: {1}
2: {1,1}
3: {3}
4: {4}
5: {4,1}
6: {4,1,1}
7: {3,4}
8: {3,4,1}
9: {3,4,1,1}
Because the sums are all different and cover {1...9}, it follows that (3,4,1,1) is counted under a(9).
The a(1) = 1 through a(9) = 9 compositions:
1 11 12 1111 113 132 1114 1133 1143
21 122 231 1222 3311 1332
111 221 111111 2221 11111111 2331
311 4111 3411
11111 1111111 11115
12222
22221
51111
111111111
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Sort[Total/@Union[ReplaceList[#, {___, s__, ___}:>{s}]]]==Range[n]&]], {n, 0, 15}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 13 2019
EXTENSIONS
a(21)-a(25) from Jinyuan Wang, Jun 26 2020
a(21)-a(25) corrected, a(26)-a(80) from Fausto A. C. Cariboni, Feb 21 2022
STATUS
approved