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 A325687 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. 17
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS A composition of n is a finite sequence of positive integers summing to n. LINKS 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}] CROSSREFS Row sums are A325676. Column k = 2 is A000027. Column k = 3 is A325688. Cf. A000079, A007318, A048004, A108917, A143823, A169942, A266223, A325592, A325680, A325685. Sequence in context: A326934 A290691 A155726 * A230079 A105400 A194516 Adjacent sequences:  A325684 A325685 A325686 * A325688 A325689 A325690 KEYWORD nonn,tabl AUTHOR Gus Wiseman, May 13 2019 STATUS approved

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Last modified June 5 18:48 EDT 2020. Contains 334854 sequences. (Running on oeis4.)