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

1, 3, 6, 7, 13, 15, 26, 27, 30, 31, 53, 55, 61, 63, 106, 107, 110, 111, 122, 123, 126, 127, 213, 215, 221, 223, 245, 247, 253, 255, 426, 427, 430, 431, 442, 443, 446, 447, 490, 491, 494, 495, 506, 507, 510, 511, 853, 855, 861, 863

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• base

Comments

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

Examples

			26 = 11010_2 has m=4, and d(i) = 1 for i=1 and 3.
53 = 110101_2 has m=5, and d(i) = 1 for i=0, 2 and 4.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..12286

Crossrefs

Cf. A126684, A000695, A062880.

Disjoint with A032937 if more than 1 digit.

Programs

Maple

F:= proc(m)
   local n0,j,S;
   n0:= 2^m + add(2^(m-1-2*j),j=0..floor((m-1)/2));
   S:= combinat[powerset]({seq(2^(m-2*j),j=1..floor(m/2))});
   map(t -> convert(t,`+`)+n0,S);
end;
`union`(seq(F(m),m=0..24)}; # Robert Israel, Mar 30 2014

Formula

a(2j+2) = 4 a(j)+3,

a(2j+1) = 4 a(j) + 2 if j <= 3*2^(m-1)-2,

a(2j+1) = 4 a(j) + 1 otherwise, where m = floor(log_2(j+1)).

Extensions

Definition corrected, incorrect cross-reference removed, and recurrence formulas by Robert Israel, Mar 30 2014