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 A095684 #11 May 01 2020 08:01:22
%S A095684 1,1,1,1,2,1,1,1,1,1,2,1,2,2,1,2,3,1,1,1,1,1,1,1,2,1,1,2,2,1,1,2,3,1,
%T A095684 2,2,2,1,2,2,3,1,2,3,3,1,2,3,4,1,1,1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,2,
%U A095684 3,1,1,2,2,2,1,1,2,2,3,1,1,2,3,3,1,1,2,3,4,1,2,2,2,2,1,2,2,2,3,1,2,2,3,3
%N A095684 Triangle read by rows. There are 2^(m-1) rows of length m, for m = 1, 2, 3, ... The rows are in lexicographic order. The rows have the property that the first entry is 1, the second distinct entry (reading from left to right) is 2, the third distinct entry is 3, etc.
%C A095684 Row k is the unique multiset that covers an initial interval of positive integers and has multiplicities equal to the parts of the k-th composition in standard order (graded reverse-lexicographic, A066099). This composition is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. For example, the 13th composition is (1,2,1), so row 13 is {1,2,2,3}. - _Gus Wiseman_, Apr 26 2020
%H A095684 J. C. Kieffer, W. Szpankowski and E.-H. Yang, <a href="https://doi.org/10.1109/TIT.2004.830747">Problems on sequences: information theory and computer science interface</a>, IEEE Trans. Inform. Theory, 50 (No. 7, 2004), 1385-1392.
%e A095684 1, 11, 12, 111, 112, 122, 123, 1111, 1112, 1122, 1123, 1222, 1223, 1233, ...
%e A095684 The 8 strings of length 4 are 1111, 1112, 1122, 1123, 1222, 1223, 1233, 1234.
%e A095684 From _Gus Wiseman_, Apr 26 2020: (Start)
%e A095684 The triangle read by columns begins:
%e A095684 1:{1} 2:{1,1} 4:{1,1,1} 8:{1,1,1,1} 16:{1,1,1,1,1}
%e A095684 3:{1,2} 5:{1,1,2} 9:{1,1,1,2} 17:{1,1,1,1,2}
%e A095684 6:{1,2,2} 10:{1,1,2,2} 18:{1,1,1,2,2}
%e A095684 7:{1,2,3} 11:{1,1,2,3} 19:{1,1,1,2,3}
%e A095684 12:{1,2,2,2} 20:{1,1,2,2,2}
%e A095684 13:{1,2,2,3} 21:{1,1,2,2,3}
%e A095684 14:{1,2,3,3} 22:{1,1,2,3,3}
%e A095684 15:{1,2,3,4} 23:{1,1,2,3,4}
%e A095684 24:{1,2,2,2,2}
%e A095684 25:{1,2,2,2,3}
%e A095684 26:{1,2,2,3,3}
%e A095684 27:{1,2,2,3,4}
%e A095684 28:{1,2,3,3,3}
%e A095684 29:{1,2,3,3,4}
%e A095684 30:{1,2,3,4,4}
%e A095684 31:{1,2,3,4,5}
%e A095684 (End)
%t A095684 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A095684 ptnToNorm[y_]:=Join@@Table[ConstantArray[i,y[[i]]],{i,Length[y]}];
%t A095684 Table[ptnToNorm[stc[n]],{n,15}] (* _Gus Wiseman_, Apr 26 2020 *)
%Y A095684 See A096299 for another version.
%Y A095684 The number of distinct parts in row n is A000120(n), also the maximum part.
%Y A095684 Row sums are A029931.
%Y A095684 Heinz numbers of rows are A057335.
%Y A095684 Row lengths are A070939.
%Y A095684 Row products are A284001.
%Y A095684 The version for prime indices is A305936.
%Y A095684 There are A333942(n) multiset partitions of row n.
%Y A095684 Multisets of compositions are counted by A034691.
%Y A095684 Combinatory separations of normal multisets are A269134.
%Y A095684 All of the following pertain to compositions in standard order (A066099):
%Y A095684 - Necklaces are A065609.
%Y A095684 - Strict compositions are A233564.
%Y A095684 - Constant compositions are A272919.
%Y A095684 - Lyndon words are A275692.
%Y A095684 - Dealings are counted by A333939.
%Y A095684 - Distinct parts are counted by A334028.
%Y A095684 Cf. A318284, A333764, A333940, A334030.
%K A095684 nonn,easy,tabf
%O A095684 1,5
%A A095684 _N. J. A. Sloane_, Jun 25 2004