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A336231 Integers whose binary digit expansion has an even number of 0’s between any two consecutive 1’s.

%I A336231 #17 Sep 22 2024 17:33:54
%S A336231 0,1,2,3,4,6,7,8,9,12,14,15,16,18,19,24,25,28,30,31,32,33,36,38,39,48,
%T A336231 50,51,56,57,60,62,63,64,66,67,72,73,76,78,79,96,97,100,102,103,112,
%U A336231 114,115,120,121,124,126,127,128,129,132,134,135,144,146,147,152
%N A336231 Integers whose binary digit expansion has an even number of 0’s between any two consecutive 1’s.
%C A336231 If m is a term then 2*m is a term too.
%C A336231 If m is an odd term and k is odd then 2^k*m+1 is a term. - _Robert Israel_, Jul 16 2020
%H A336231 Robert Israel, <a href="/A336231/b336231.txt">Table of n, a(n) for n = 1..10000</a>
%H A336231 Daniel Glasscock, Joel Moreira, and Florian K. Richter, <a href="https://arxiv.org/abs/2007.05480">Additive transversality of fractal sets in the reals and the integers</a>, arXiv:2007.05480 [math.NT], 2020. See Aeven p. 34.
%e A336231 9 is 1001 in binary, with 2 (an even number) consecutive zeros, so 9 is a term.
%p A336231 B[1]:= {1}: S[0]:= {0}: S[1]:= {1}: count:= 2:
%p A336231 for d from 2 while count < 200 do
%p A336231   B[d]:= map(op, {seq(map(t -> t*2^k+1, B[d-k]), k=1..d-1,2)});
%p A336231   S[d]:= B[d] union map(`*`, S[d-1], 2);
%p A336231   count:= count+nops(S[d]);
%p A336231 od:
%p A336231 [seq(op(sort(convert(S[t], list))), t=0..d-1)]; # _Robert Israel_, Jul 16 2020
%o A336231 (PARI) isok(n) = {my(vpos = select(x->(x==1), binary(n), 1)); for (i=1, #vpos-1, if ((vpos[i+1]-vpos[i]-1) % 2, return (0));); return(1);}
%Y A336231 Cf. A007088, A060142, A336232.
%K A336231 nonn,base
%O A336231 1,3
%A A336231 _Michel Marcus_, Jul 13 2020