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A329867
Runs-resistance minus cuts-resistance of the binary expansion of n.
7
0, -1, 1, -1, 1, 1, 1, -2, 0, 1, 1, 2, 0, 2, 0, -3, -1, 0, 3, 2, 2, 1, 3, 1, 0, 2, 2, 0, 0, 1, -1, -4, -2, -1, 2, 0, 0, 3, 2, 0, 1, 3, 1, 2, 1, 2, 2, 0, -1, 0, 1, 0, 2, 2, 0, -1, -1, 0, 1, -1, -1, 0, -2, -5, -3, -2, 1, -1, -1, 2, 0, 1, -1, 0, 3, 4, 2, 3, 0
OFFSET
0,8
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
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.
FORMULA
For n > 1, a(2^n) = 3 - n.
For n > 1, a(2^n - 1) = 1 - n.
EXAMPLE
The sequence of binary expansions together with their runs-resistances and cuts-resistances, and their differences, begins:
0 (): 0 - 0 = 0
1 (1): 0 - 1 = -1
2 (10): 2 - 1 = 1
3 (11): 1 - 2 = -1
4 (100): 3 - 2 = 1
5 (101): 2 - 1 = 1
6 (110): 3 - 2 = 1
7 (111): 1 - 3 = -2
8 (1000): 3 - 3 = 0
9 (1001): 3 - 2 = 1
10 (1010): 2 - 1 = 1
11 (1011): 4 - 2 = 2
12 (1100): 2 - 2 = 0
13 (1101): 4 - 2 = 2
14 (1110): 3 - 3 = 0
15 (1111): 1 - 4 = -3
16 (10000): 3 - 4 = -1
17 (10001): 3 - 3 = 0
18 (10010): 5 - 2 = 3
19 (10011): 4 - 2 = 2
20 (10100): 4 - 2 = 2
MATHEMATICA
runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
Table[If[n==0, 0, runsres[IntegerDigits[n, 2]]-degdep[IntegerDigits[n, 2]]], {n, 0, 100}]
CROSSREFS
Positions of 0's are A329865.
Positions of -1's are A329866.
Sorted positions of first appearances are A329868.
Compositions with runs-resistance equal to cuts-resistance are A329864.
Compositions with runs-resistance = cuts-resistance minus 1 are A329869.
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.
Sequence in context: A294891 A294879 A085122 * A342652 A083715 A037135
KEYWORD
sign
AUTHOR
Gus Wiseman, Nov 23 2019
STATUS
approved