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Numbers with no 11 or 000 in their binary expansion.
3

%I #20 Jul 21 2021 09:05:57

%S 0,1,2,4,5,9,10,18,20,21,36,37,41,42,73,74,82,84,85,146,148,149,164,

%T 165,169,170,292,293,297,298,329,330,338,340,341,585,586,594,596,597,

%U 658,660,661,676,677,681,682,1170,1172,1173

%N Numbers with no 11 or 000 in their binary expansion.

%C The number of n-bit numbers in this sequence for n>1 is given by a(n+6) where a is the Padovan sequence A000931.

%H Chai Wah Wu, <a href="https://arxiv.org/abs/1810.02293">Record values in appending and prepending bitstrings to runs of binary digits</a>, arXiv:1810.02293 [math.NT], 2018.

%t Select[Range[0, 1200], And[AllTrue[#1, # < 2 &], AllTrue[#2, # < 3 &]] & @@ {Part[#, 2 Range@ Ceiling[Length[#]/2] - 1], Part[#, 2 Range@ Floor[Length[#]/2]]} &@ Map[Length, Split@ IntegerDigits[#, 2]] &] (* _Michael De Vlieger_, Dec 26 2018 *)

%t Select[Range[0,1200],SequenceCount[IntegerDigits[#,2],{1,1}] == SequenceCount[ IntegerDigits[ #,2],{0,0,0}]==0&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, May 16 2019 *)

%o (Python)

%o def A086638():

%o yield 0

%o for x in A086638():

%o if x & 3:

%o yield 2*x

%o if not (x & 1):

%o yield 2*x + 1

%o a = A086638(); print([next(a) for _ in range(100)])

%Y Cf. A000931.

%K easy,base,nonn

%O 0,3

%A _David Eppstein_, Sep 14 2003