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A385817 Irregular triangle read by rows listing the lengths of maximal runs (sequences of consecutive elements increasing by 1) of binary indices, duplicate rows removed.

%I A385817 #10 Jul 19 2025 10:14:08
%S A385817 1,2,1,1,3,2,1,1,2,4,1,1,1,3,1,2,2,1,3,5,2,1,1,1,2,1,4,1,1,1,2,3,2,2,
%T A385817 3,1,4,6,1,1,1,1,3,1,1,2,2,1,1,3,1,5,1,2,1,2,1,2,2,4,2,1,1,3,3,3,2,4,
%U A385817 1,5,7,2,1,1,1,1,2,1,1,4,1,1,1,1,2,1
%N A385817 Irregular triangle read by rows listing the lengths of maximal runs (sequences of consecutive elements increasing by 1) of binary indices, duplicate rows removed.
%C A385817 A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
%C A385817 This is the triangle A245563, except all duplicates after the first instance of each composition are removed. It lists all compositions in order of their first appearance as a row of A245563.
%e A385817 The binary indices of 53 are {1,3,5,6}, with maximal runs ((1),(3),(5,6)), with lengths (1,1,2). After removing duplicates, this is our row 16.
%e A385817 Triangle begins:
%e A385817    0: .
%e A385817    1: 1
%e A385817    2: 2
%e A385817    3: 1 1
%e A385817    4: 3
%e A385817    5: 2 1
%e A385817    6: 1 2
%e A385817    7: 4
%e A385817    8: 1 1 1
%e A385817    9: 3 1
%e A385817   10: 2 2
%e A385817   11: 1 3
%e A385817   12: 5
%e A385817   13: 2 1 1
%e A385817   14: 1 2 1
%e A385817   15: 4 1
%e A385817   16: 1 1 2
%e A385817   17: 3 2
%e A385817   18: 2 3
%e A385817   19: 1 4
%e A385817   20: 6
%e A385817   21: 1 1 1 1
%t A385817 DeleteDuplicates[Table[Length/@Split[Join@@Position[Reverse[IntegerDigits[n,2]],1],#2==#1+1&],{n,0,100}]]
%Y A385817 In the following references, "before" is short for "before removing duplicate rows".
%Y A385817 Positions of singleton rows appear to be A000071 = A000045-1, before A023758.
%Y A385817 Positions of firsts appearances appear to be A001629.
%Y A385817 Positions of rows of the form (1,1,...) appear to be A055588 = A001906+1.
%Y A385817 First term of each row appears to be A083368.
%Y A385817 Row sums appear to be A200648, before A000120.
%Y A385817 Row lengths after the first row appear to be A200650+1, before A069010 = A037800+1.
%Y A385817 Before the removals we had A245563 (except first term), see A245562, A246029, A328592.
%Y A385817 For anti-run ranks we have A385816, before A348366, firsts A052499.
%Y A385817 Standard composition numbers of rows are A385818, before A385889.
%Y A385817 For anti-runs we have A385886, before A384877, firsts A384878.
%Y A385817 Cf. A044813, A048793, A164707, A200649, A242882, A243815, A384175, A384890.
%K A385817 nonn,tabf
%O A385817 0,2
%A A385817 _Gus Wiseman_, Jul 14 2025