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A033053 Numbers whose base-2 representation Sum_{i=0..m} d(i)*2^i has d(i)=1 when i != m mod 2.

%I A033053 #40 Jul 23 2025 05:45:29
%S A033053 1,3,6,7,13,15,26,27,30,31,53,55,61,63,106,107,110,111,122,123,126,
%T A033053 127,213,215,221,223,245,247,253,255,426,427,430,431,442,443,446,447,
%U A033053 490,491,494,495,506,507,510,511,853,855,861,863
%N A033053 Numbers whose base-2 representation Sum_{i=0..m} d(i)*2^i has d(i)=1 when i != m mod 2.
%C A033053 Numbers 2^(2k)-1 - A062880(m) where 2^(2k-2) >= A062880(m) or 2^(2k+1)-1 - A000695(m) where 2^(2k-1) >= A000695(m). - _Franklin T. Adams-Watters_, Aug 30 2014
%H A033053 Robert Israel, <a href="/A033053/b033053.txt">Table of n, a(n) for n = 1..12286</a>
%F A033053 a(2j+2) = 4 a(j)+3,
%F A033053 a(2j+1) = 4 a(j) + 2 if j <= 3*2^(m-1)-2,
%F A033053 a(2j+1) = 4 a(j) + 1 otherwise, where m = floor(log_2(j+1)).
%e A033053 26 = 11010_2 has m=4, and d(i) = 1 for i=1 and 3.
%e A033053 53 = 110101_2 has m=5, and d(i) = 1 for i=0, 2 and 4.
%p A033053 F:= proc(m)
%p A033053    local n0,j,S;
%p A033053    n0:= 2^m + add(2^(m-1-2*j),j=0..floor((m-1)/2));
%p A033053    S:= combinat[powerset]({seq(2^(m-2*j),j=1..floor(m/2))});
%p A033053    map(t -> convert(t,`+`)+n0,S);
%p A033053 end;
%p A033053 `union`(seq(F(m),m=0..24)}; # _Robert Israel_, Mar 30 2014
%Y A033053 Cf. A126684, A000695, A062880.
%Y A033053 Disjoint with A032937 if more than 1 digit.
%K A033053 nonn,base
%O A033053 1,2
%A A033053 _Clark Kimberling_
%E A033053 Definition corrected, incorrect cross-reference removed, and recurrence formulas by _Robert Israel_, Mar 30 2014