This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A353850 #29 Aug 09 2023 11:47:51
%S A353850 1,1,2,4,5,12,24,38,52,111,218,286,520,792,1358,2628,4155,5508,9246,
%T A353850 13182,23480,45150,54540,94986,146016,213725,301104,478586,851506,
%U A353850 1302234,1775482,2696942,3746894,6077784,8194466,12638334,21763463,28423976,45309850,62955524,94345474
%N A353850 Number of integer compositions of n with all distinct run-sums.
%C A353850 Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).
%H A353850 Joseph Likar, <a href="/A353850/b353850.txt">Table of n, a(n) for n = 0..120</a>
%e A353850 The a(0) = 1 through a(5) = 12 compositions:
%e A353850 () (1) (2) (3) (4) (5)
%e A353850 (11) (12) (13) (14)
%e A353850 (21) (22) (23)
%e A353850 (111) (31) (32)
%e A353850 (1111) (41)
%e A353850 (113)
%e A353850 (122)
%e A353850 (221)
%e A353850 (311)
%e A353850 (1112)
%e A353850 (2111)
%e A353850 (11111)
%e A353850 For n=4, (211) is invalid because the two runs (2) and (11) have the same sum. - _Joseph Likar_, Aug 04 2023
%t A353850 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], UnsameQ@@Total/@Split[#]&]],{n,0,15}]
%Y A353850 For distinct parts instead of run-sums we have A032020.
%Y A353850 For distinct multiplicities instead of run-sums we have A242882.
%Y A353850 For distinct run-lengths instead of run-sums we have A329739, ptns A098859.
%Y A353850 For runs instead of run-sums we have A351013.
%Y A353850 For partitions we have A353837, ranked by A353838 (complement A353839).
%Y A353850 For equal instead of distinct run-sums we have A353851, ptns A304442.
%Y A353850 These compositions are ranked by A353852.
%Y A353850 The weak version (rucksack compositions) is A354580, ranked by A354581.
%Y A353850 A003242 counts anti-run compositions, ranked by A333489.
%Y A353850 A005811 counts runs in binary expansion.
%Y A353850 A011782 counts compositions.
%Y A353850 A175413 lists numbers whose binary expansion has all distinct runs.
%Y A353850 A351014 counts distinct runs in standard compositions, firsts A351015.
%Y A353850 A353847 gives composition run-sum transformation.
%Y A353850 A353929 counts distinct runs in binary expansion, firsts A353930.
%Y A353850 Cf. A238279, A333755, A351016, A351017, A353832, A353848, A353849, A353853-A353859, A353860, A353863, A353932.
%K A353850 nonn
%O A353850 0,3
%A A353850 _Gus Wiseman_, May 31 2022
%E A353850 Terms a(21) and onwards from _Joseph Likar_, Aug 04 2023