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Runs-resistance of binary representation of n.
65

%I #27 Jul 01 2020 23:44:55

%S 1,2,1,3,2,3,1,3,3,2,4,2,4,3,1,3,3,5,4,4,2,5,4,3,4,4,3,3,4,3,1,3,3,5,

%T 3,3,5,4,3,4,5,2,4,3,4,5,4,3,3,3,2,4,4,3,3,2,3,4,3,3,4,3,1,3,3,5,3,3,

%U 5,3,4,3,3,5,6,4,5,3,3,4,5,4,4,4,2,5,4,5,5,4,5,5,4,5,4

%N Runs-resistance of binary representation of n.

%C Following Lenormand (2003), we define the "runs-resistance" of a finite list L to be the number of times the RUNS transformation must be applied to L in order to reduce L to a list with a single element.

%C Here it is immaterial whether we read the binary representation of n from left to right or right to left.

%C The RUNS transformation must be applied at least once, in order to obtain a list, so a(n) >= 1.

%H N. J. A. Sloane, <a href="/A318928/b318928.txt">Table of n, a(n) for n = 1..10000</a>

%H Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%e 11 in binary is [1, 0, 1, 1],

%e which has runs of lengths [1, 1, 2],

%e which has runs of lengths [2, 1],

%e which has runs of lengths [1, 1],

%e which has a single run of length [2].

%e This took four steps, so a(11) = 4.

%p with(transforms);

%p # compute Lenormand's "resistance" of a list

%p resist:=proc(a) local ct,i,b;

%p if whattype(a) <> list then ERROR("input must be a list"); fi:

%p ct:=0; b:=a; for i from 1 to 100 do

%p if nops(b)=1 then return(ct); fi;

%p b:=RUNS(b); ct:=ct+1; od; end;

%p a:=[1];

%p for n from 2 to 100 do

%p b:=convert(n,base,2);

%p r:=resist(b);

%p a:=[op(a),r];

%p od:

%t Table[If[n == 1, 1, Length[NestWhileList[Length/@Split[#] &, IntegerDigits[n, 2], Length[#] > 1 &]] - 1], {n, 50}] (* _Gus Wiseman_, Nov 25 2019 *)

%Y See A319103 for an inverse, and A319417 and A319418 for records.

%Y Ignoring the first digit gives A329870.

%Y Cuts-resistance is A319416.

%Y Compositions counted by runs-resistance are A329744.

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

%Y Cf. A107907, A319420, A329860, A329861, A329865, A329867.

%K nonn,base,nice

%O 1,2

%A _N. J. A. Sloane_, Sep 09 2018

%E a(1) corrected by _N. J. A. Sloane_, Sep 20 2018