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A325685
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Number of compositions of n whose distinct consecutive subsequences have different sums, and such that these sums cover an initial interval of positive integers.
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12
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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
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OFFSET
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0,4
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COMMENTS
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A composition of n is a finite sequence of positive integers summing to n.
Compare to the definition of perfect partitions (A002033).
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Sort[Total/@Union[ReplaceList[#, {___, s__, ___}:>{s}]]]==Range[n]&]], {n, 0, 15}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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