A329868 - OEIS

A329868

Sorted positions of first appearances in A329867 (difference between the runs-resistance and the cuts-resistance of binary expansion) of each element in the image.

0, 1, 2, 7, 11, 15, 18, 31, 63, 75, 127, 255, 511, 1023, 1234, 2047, 4095, 8191, 9638, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

LINKS

Table of n, a(n) for n=1..29.

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

EXAMPLE

The sequence of terms together with their binary expansions begins:

1: 1

2: 10

7: 111

11: 1011

15: 1111

18: 10010

31: 11111

63: 111111

75: 1001011

127: 1111111

255: 11111111

511: 111111111

1023: 1111111111

1234: 10011010010

2047: 11111111111

4095: 111111111111

8191: 1111111111111

9638: 10010110100110

16383: 11111111111111

32767: 111111111111111

65535: 1111111111111111

MATHEMATICA

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;

das=Table[If[n==0, 0, runsres[IntegerDigits[n, 2]]-degdep[IntegerDigits[n, 2]]], {n, 0, 1000000}];

Table[Position[das, i][[1, 1]]-1, {i, First/@Gather[das]}]

CROSSREFS

Sorted positions of first appearances in A329867.

Compositions with runs-resistance equal to cuts-resistance are A329864.

Runs-resistance of binary expansion is A318928.

Cuts-resistance of binary expansion is A319416.

Compositions counted by runs-resistance are A329744.

Compositions counted by cuts-resistance are A329861.

Binary words counted by runs-resistance are A319411 and A329767.

Binary words counted by cuts-resistance are A319421 and A329860.

Cf. A000975, A003242, A107907, A164707, A329738, A329865, A329866.

Sequence in context: A287107 A308550 A018308 * A184485 A001960 A174008

Adjacent sequences: A329865 A329866 A329867 * A329869 A329870 A329871

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Nov 23 2019

STATUS

approved