%I #8 Nov 24 2019 10:00:35

%S 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,

%T -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,

%U 0,-2,-5,-3,-2,1,-1,-1,2,0,1,-1,0,3,4,2,3,0

%N Runs-resistance minus cuts-resistance of the binary expansion of n.

%C 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.

%C 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.

%H Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.

%F For n > 1, a(2^n) = 3 - n.

%F For n > 1, a(2^n - 1) = 1 - n.

%e The sequence of binary expansions together with their runs-resistances and cuts-resistances, and their differences, begins:

%e 0 (): 0 - 0 = 0

%e 1 (1): 0 - 1 = -1

%e 2 (10): 2 - 1 = 1

%e 3 (11): 1 - 2 = -1

%e 4 (100): 3 - 2 = 1

%e 5 (101): 2 - 1 = 1

%e 6 (110): 3 - 2 = 1

%e 7 (111): 1 - 3 = -2

%e 8 (1000): 3 - 3 = 0

%e 9 (1001): 3 - 2 = 1

%e 10 (1010): 2 - 1 = 1

%e 11 (1011): 4 - 2 = 2

%e 12 (1100): 2 - 2 = 0

%e 13 (1101): 4 - 2 = 2

%e 14 (1110): 3 - 3 = 0

%e 15 (1111): 1 - 4 = -3

%e 16 (10000): 3 - 4 = -1

%e 17 (10001): 3 - 3 = 0

%e 18 (10010): 5 - 2 = 3

%e 19 (10011): 4 - 2 = 2

%e 20 (10100): 4 - 2 = 2

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

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

%t Table[If[n==0,0,runsres[IntegerDigits[n,2]]-degdep[IntegerDigits[n,2]]],{n,0,100}]

%Y Positions of 0's are A329865.

%Y Positions of -1's are A329866.

%Y Sorted positions of first appearances are A329868.

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

%Y Compositions with runs-resistance = cuts-resistance minus 1 are A329869.

%Y Runs-resistance of binary expansion is A318928.

%Y Cuts-resistance of binary expansion is A319416.

%Y Compositions counted by runs-resistance are A329744.

%Y Compositions counted by cuts-resistance are A329861.

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

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

%Y Cf. A000975, A003242, A107907, A164707, A329738.

%K sign

%O 0,8

%A _Gus Wiseman_, Nov 23 2019