A384877 - OEIS

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.

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, 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, 2, 3, 1, 2, 1, 1, 2, 2, 3, 3, 1, 3, 1, 2, 2, 2

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OFFSET

0,6

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..83.

EXAMPLE

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).

Triangle begins:

0: ()

1: (1)

2: (1)

3: (1,1)

4: (1)

5: (2)

6: (1,1)

7: (1,1,1)

8: (1)

9: (2)

10: (2)

11: (1,2)

12: (1,1)

13: (2,1)

14: (1,1,1)

15: (1,1,1,1)

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

Table[Length/@Split[bpe[n], #2!=#1+1&], {n, 0, 100}]

CROSSREFS

Row-sums are A000120.

Positions of rows of the form (1,1,...) are A023758.

Positions of first appearances of each distinct row appear to be A052499.

For runs instead of anti-runs we have A245563, reverse A245562.

Row-lengths are A384890.

A355394 counts partitions without a neighborless part, singleton case A355393.

A356606 counts strict partitions without a neighborless part, complement A356607.

A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.

Cf. A044813, A048793, A069010, A164707, A243815, A246029, A328592, A384177, A384877, A384879, A384893.

Sequence in context: A359997 A122172 A030613 * A025910 A002637 A166279

Adjacent sequences: A384874 A384875 A384876 * A384878 A384879 A384880

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Jun 17 2025

STATUS

approved