OFFSET

1,2

COMMENTS

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

LINKS

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

EXAMPLE

The terms, binary expansions, and standard compositions begin:

1: 1 (1)

2: 10 (2)

11: 1011 (2,1,1)

183: 10110111 (2,1,2,1,1,1)

5871: 1011011101111 (2,1,2,1,1,2,1,1,1,1)

375775: 1011011101111011111 (2,1,2,1,1,2,1,1,1,2,1,1,1,1,1)

MATHEMATICA

qe=Table[Length[Union[Total/@Split[IntegerDigits[n, 2]]]], {n, 1, 10000}];

Table[Position[qe, i][[1, 1]], {i, Max@@qe}]

PROG

(PARI) a(n) = {my(t=1); if(n==2, t<<=1, for(k=3, n, t = (t<<k) + (2^(k-1)-1))); t} \\ Andrew Howroyd, Jan 01 2023

CROSSREFS

Essentially the same as A215203.

For prime indices instead of binary expansion we have A006939.

Numbers whose binary expansion has all distinct runs are A175413.

These are the positions of first appearances in A353929.

A005811 counts runs in binary expansion.

A242882 counts compositions with distinct multiplicities.

A318928 gives runs-resistance of binary expansion.

A351014 counts distinct runs in standard compositions.

A353864 counts rucksack partitions.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 07 2022

EXTENSIONS

Offset corrected and terms a(7) and beyond from Andrew Howroyd, Jan 01 2023

STATUS

approved