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A325687
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Triangle read by rows where T(n,k) is the number of length-k compositions of n such that every distinct consecutive subsequence has a different sum.
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19
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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
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OFFSET
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1,5
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COMMENTS
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A composition of n is a finite sequence of positive integers summing to n.
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n, {k}], UnsameQ@@Total/@Union[ReplaceList[#, {___, s__, ___}:>{s}]]&]], {n, 15}, {k, n}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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