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A325685 Number of compositions of n whose distinct consecutive subsequences have different sums, and such that these sums cover an initial interval of positive integers. 12
1, 1, 1, 3, 1, 5, 3, 5, 3, 9, 1, 9, 5, 7, 5, 11, 1, 13, 5, 9, 5, 13, 3, 13, 7, 9, 5, 17, 1, 17, 5, 9, 9, 15, 5, 15, 5, 13, 5, 21, 1, 17, 9, 9, 9, 17, 3, 21, 7, 13, 5, 17, 5, 21, 9, 13, 5, 21, 1, 21, 9, 11, 13, 19, 5, 17, 5, 17, 5, 29, 1, 21, 9, 9, 13, 17, 5, 25, 7, 17, 7 (list; graph; refs; listen; history; text; internal format)
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
Sequence in context: A143865 A071168 A091926 * A109606 A318727 A307806
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

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)