cp's OEIS Frontend

A384877 Irregular triangle read by rows where row k lists the lengths of maximal anti-runs (increasing by more than 1) in the binary indices of n.

%I A384877 #9 Jun 18 2025 07:35:50
%S A384877 1,1,1,1,1,2,1,1,1,1,1,1,2,2,1,2,1,1,2,1,1,1,1,1,1,1,1,1,2,2,1,2,2,3,
%T A384877 1,2,1,1,2,1,1,2,1,2,1,1,2,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,2,1,2,
%U A384877 2,3,1,2,1,1,2,2,3,3,1,3,1,2,2,2
%N A384877 Irregular triangle read by rows where row k lists the lengths of maximal anti-runs (increasing by more than 1) in the binary indices of n.
%C A384877 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 A384877 The binary indices of 182 are {2,3,5,6,8}, with maximal anti-runs ((2),(3,5),(6,8)) so row 182 is (1,2,2).
%e A384877 Triangle begins:
%e A384877    0: ()
%e A384877    1: (1)
%e A384877    2: (1)
%e A384877    3: (1,1)
%e A384877    4: (1)
%e A384877    5: (2)
%e A384877    6: (1,1)
%e A384877    7: (1,1,1)
%e A384877    8: (1)
%e A384877    9: (2)
%e A384877   10: (2)
%e A384877   11: (1,2)
%e A384877   12: (1,1)
%e A384877   13: (2,1)
%e A384877   14: (1,1,1)
%e A384877   15: (1,1,1,1)
%t A384877 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A384877 Table[Length/@Split[bpe[n],#2!=#1+1&],{n,0,100}]
%Y A384877 Row-sums are A000120.
%Y A384877 Positions of rows of the form (1,1,...) are A023758.
%Y A384877 Positions of first appearances of each distinct row appear to be A052499.
%Y A384877 For runs instead of anti-runs we have A245563, reverse A245562.
%Y A384877 Row-lengths are A384890.
%Y A384877 A355394 counts partitions without a neighborless part, singleton case A355393.
%Y A384877 A356606 counts strict partitions without a neighborless part, complement A356607.
%Y A384877 A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
%Y A384877 Cf. A044813, A048793, A069010, A164707, A243815, A246029, A328592, A384177, A384877, A384879, A384893.
%K A384877 nonn,tabf
%O A384877 0,6
%A A384877 _Gus Wiseman_, Jun 17 2025