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%I A329867 #8 Nov 24 2019 10:00:35
%S A329867 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 A329867 -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 A329867 0,-2,-5,-3,-2,1,-1,-1,2,0,1,-1,0,3,4,2,3,0
%N A329867 Runs-resistance minus cuts-resistance of the binary expansion of n.
%C A329867 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 A329867 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 A329867 Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%F A329867 For n > 1, a(2^n) = 3 - n.
%F A329867 For n > 1, a(2^n - 1) = 1 - n.
%e A329867 The sequence of binary expansions together with their runs-resistances and cuts-resistances, and their differences, begins:
%e A329867    0      (): 0 - 0 =  0
%e A329867    1     (1): 0 - 1 = -1
%e A329867    2    (10): 2 - 1 =  1
%e A329867    3    (11): 1 - 2 = -1
%e A329867    4   (100): 3 - 2 =  1
%e A329867    5   (101): 2 - 1 =  1
%e A329867    6   (110): 3 - 2 =  1
%e A329867    7   (111): 1 - 3 = -2
%e A329867    8  (1000): 3 - 3 =  0
%e A329867    9  (1001): 3 - 2 =  1
%e A329867   10  (1010): 2 - 1 =  1
%e A329867   11  (1011): 4 - 2 =  2
%e A329867   12  (1100): 2 - 2 =  0
%e A329867   13  (1101): 4 - 2 =  2
%e A329867   14  (1110): 3 - 3 =  0
%e A329867   15  (1111): 1 - 4 = -3
%e A329867   16 (10000): 3 - 4 = -1
%e A329867   17 (10001): 3 - 3 =  0
%e A329867   18 (10010): 5 - 2 =  3
%e A329867   19 (10011): 4 - 2 =  2
%e A329867   20 (10100): 4 - 2 =  2
%t A329867 runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
%t A329867 degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
%t A329867 Table[If[n==0,0,runsres[IntegerDigits[n,2]]-degdep[IntegerDigits[n,2]]],{n,0,100}]
%Y A329867 Positions of 0's are A329865.
%Y A329867 Positions of -1's are A329866.
%Y A329867 Sorted positions of first appearances are A329868.
%Y A329867 Compositions with runs-resistance equal to cuts-resistance are A329864.
%Y A329867 Compositions with runs-resistance = cuts-resistance minus 1 are A329869.
%Y A329867 Runs-resistance of binary expansion is A318928.
%Y A329867 Cuts-resistance of binary expansion is A319416.
%Y A329867 Compositions counted by runs-resistance are A329744.
%Y A329867 Compositions counted by cuts-resistance are A329861.
%Y A329867 Binary words counted by runs-resistance are A319411 and A329767.
%Y A329867 Binary words counted by cuts-resistance are A319421 and A329860.
%Y A329867 Cf. A000975, A003242, A107907, A164707, A329738.
%K A329867 sign
%O A329867 0,8
%A A329867 _Gus Wiseman_, Nov 23 2019

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