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

1,2

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

Links

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

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)).

Example

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.

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

Crossrefs

Cf. A126684, A000695, A062880.

Sequence in context: A176301 A191290 A137595 * A248388 A354765 A107850

Adjacent sequences: A033050 A033051 A033052 * A033054 A033055 A033056

Keyword

nonn,base

Author

Clark Kimberling

Extensions

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

Status

approved