A107750 If n=0 then 0, else smallest number greater than its predecessor and having either more or fewer zeros in its binary representation.

0, 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92, 93

Offset: 0

Author

Reinhard Zumkeller, May 23 2005

Keywords

• nonn

Comments

Essentially the complement of A016825 with respect to the nonnegative integers (except for 2). A023416(a(n+1)) <> A023416(a(n)).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A107686, A007088, A023416.

Programs

Haskell

a107750 n = a107750_list !! n
a107750_list = 0 : f 0 where
   f x = y : f y where
     y = head [z | z <- [x + 1 ..], a023416 z /= a023416 x]
-- Reinhard Zumkeller, Jul 07 2014

Mathematica

Table[n - Sign[Floor[n/3]] + Floor[(1/2) Sum[Ceiling[(i + 2)/3] - Floor[(i + 2)/3], {i, n}]], {n, 0, 50}] (* Wesley Ivan Hurt, Jun 16 2014 *)

Formula

a(n+1) = a(n) + A107751(n).

For k >= 0, 0 <= i <= 3*2^k:

a(6*2^k + i) = a(3*2^k + i) + 4*2^k,

a(9*2^k + i) = a(3*2^k + i) + 8*2^k.

a(n) = n - sign(floor(n/3)) + floor( (1/2)*sum_{i=1..n} ( ceiling((i+2)/3) - floor((i+2)/3) ) ). - Wesley Ivan Hurt, Jun 16 2014

Conjectures from Colin Barker, Jul 24 2017: (Start)

G.f.: x*(1+x)*(1+x^2-x^3+x^4) / ((1-x)^2*(1+x+x^2)).

a(n) = a(n-1) + a(n-3) - a(n-4) for n>3.

(End)