cp's OEIS Frontend

A384878 Position of first appearance of n in the flattened version of the triangle A384877, whose m-th row lists the lengths of maximal anti-runs in the binary indices of m.

%I A384878 #12 Jun 25 2025 18:04:30
%S A384878 1,6,34,178,882,4210,19570,89202,400498,1776754
%N A384878 Position of first appearance of n in the flattened version of the triangle A384877, whose m-th row lists the lengths of maximal anti-runs in the binary indices of m.
%C A384878 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.
%e A384878 The set of binary indices of each nonnegative integer and its partition into anti-runs begins:
%e A384878   0: {}      {{}}
%e A384878   1: {1}     {{1}}
%e A384878   2: {2}     {{2}}
%e A384878   3: {1,2}   {{1},{2}}
%e A384878   4: {3}     {{3}}
%e A384878   5: {1,3}   {{1,3}}
%e A384878   6: {2,3}   {{2},{3}}
%e A384878   7: {1,2,3} {{1},{2},{3}}
%e A384878 The flattened version begins: {}, {1}, {2}, {1}, {2}, {3}, {1,3}, {2}, {3}, {1}, {2}, {3}. Of these sets, the first of length 2 is the sixth (starting with 0), so we have a(2) = 6.
%t A384878 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A384878 mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
%t A384878 q=Join@@Table[Length/@Split[bpe[n],#2!=#1+1&],{n,0,100}];
%t A384878 Table[Position[q,i][[1,1]],{i,mnrm[q]}]
%Y A384878 For runs instead of anti-runs we have A001792.
%Y A384878 The unflattened version is A052499.
%Y A384878 Positions of first appearances in A384877, see A000120, A245562, A245563, A384890.
%Y A384878 A023758 lists differences of powers of 2.
%Y A384878 A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
%Y A384878 Cf. A044813, A048793, A069010, A384879, A384893.
%K A384878 nonn,more
%O A384878 1,2
%A A384878 _Gus Wiseman_, Jun 23 2025