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Numbers whose binary expansion has cuts-resistance <= 2.
1

%I #10 Nov 29 2019 18:26:07

%S 0,1,2,3,4,5,6,9,10,11,12,13,18,19,20,21,22,25,26,37,38,41,42,43,44,

%T 45,50,51,52,53,74,75,76,77,82,83,84,85,86,89,90,101,102,105,106,149,

%U 150,153,154,165,166,169,170,171,172,173,178,179,180,181,202,203

%N Numbers whose binary expansion has cuts-resistance <= 2.

%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.

%C Also numbers whose binary expansion is a balanced word (see A027383 for definition).

%C Also numbers whose binary expansion has all run-lengths 1 or 2 and whose sequence of run-lengths has no odd-length run of 1's sandwiched between two 2's.

%e The sequence of terms together with their binary expansions begins:

%e 0:

%e 1: 1

%e 2: 10

%e 3: 11

%e 4: 100

%e 5: 101

%e 6: 110

%e 9: 1001

%e 10: 1010

%e 11: 1011

%e 12: 1100

%e 13: 1101

%e 18: 10010

%e 19: 10011

%e 20: 10100

%e 21: 10101

%e 22: 10110

%e 25: 11001

%e 26: 11010

%e 37: 100101

%e 38: 100110

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

%t Select[Range[0,100],degdep[IntegerDigits[#,2]]<=2&]

%Y Union of A000975 and A329862.

%Y Balanced binary words are counted by A027383.

%Y Compositions with cuts-resistance <= 2 are A330028.

%Y Cuts-resistance of binary expansion is A319416.

%Y Cf. A027383, A098504, A107907, A164707, A329860, A329861, A329863, A329865.

%K nonn

%O 1,3

%A _Gus Wiseman_, Nov 27 2019