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%I #23 Feb 12 2022 10:42:28
%S 1,1,2,4,5,10,12,24,26,47,50,96,104,172,188,322,335,552,590,938,1002,
%T 1612,1648,2586,2862,4131,4418,6718,7122,10332,11166,15930,17446,
%U 24834,26166,37146,41087,55732,59592,84068,89740,122106,133070,177876,194024,262840,278626
%N Number of compositions of n such that every distinct consecutive subsequence has a different sum.
%C A composition of n is a finite sequence of positive integers summing to n.
%C Compare to the definition of knapsack partitions (A108917).
%H Fausto A. C. Cariboni, <a href="/A325676/b325676.txt">Table of n, a(n) for n = 0..100</a>
%e The distinct consecutive subsequences of (1,4,4,3) together with their sums are:
%e 1: {1}
%e 3: {3}
%e 4: {4}
%e 5: {1,4}
%e 7: {4,3}
%e 8: {4,4}
%e 9: {1,4,4}
%e 11: {4,4,3}
%e 12: {1,4,4,3}
%e Because the sums are all different, (1,4,4,3) is counted under a(12).
%e The a(1) = 1 through a(6) = 12 compositions:
%e (1) (2) (3) (4) (5) (6)
%e (11) (12) (13) (14) (15)
%e (21) (22) (23) (24)
%e (111) (31) (32) (33)
%e (1111) (41) (42)
%e (113) (51)
%e (122) (114)
%e (221) (132)
%e (311) (222)
%e (11111) (231)
%e (411)
%e (111111)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@Union[ReplaceList[#,{___,s__,___}:>{s}]]&]],{n,0,15}]
%Y Cf. A000079, A103295, A108917, A169942, A235998, A321143.
%Y Cf. A325466, A325545, A325680, A325682, A325685, A325687, A325688.
%K nonn
%O 0,3
%A _Gus Wiseman_, May 13 2019
%E a(21)-a(22) from _Jinyuan Wang_, Jun 20 2020
%E a(23)-a(25) from _Robert Price_, Jun 19 2021
%E a(26)-a(46) from _Fausto A. C. Cariboni_, Feb 10 2022
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