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A329867
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Runs-resistance minus cuts-resistance of the binary expansion of n.
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7
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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
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OFFSET
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0,8
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COMMENTS
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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.
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LINKS
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FORMULA
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For n > 1, a(2^n) = 3 - n.
For n > 1, a(2^n - 1) = 1 - n.
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EXAMPLE
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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
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MATHEMATICA
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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}]
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CROSSREFS
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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.
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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