0,4

Equals A103530(n+2) - 1. - Philippe DELEHAM, Apr 06 2005

David Applegate, Benoit Cloitre, Philippe DELEHAM and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

David Applegate, Benoit Cloitre, Philippe DELEHAM and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].

a(n) = n - Sum_{ k >= 0, 2^{k+1} <= n, n == k mod 2^(k+1) } 2^(k+1).

Structure: blocks of size 2^k taken from A105025, interspersed with terms a(n) itself! Thus a(2^k + k - 1 ) = a(k-1) for k >= 1.

Remarks from David Applegate (david(AT)research.att.com), Apr 06 2005: "a(n) = 2^k + a(n-2^k) if k >= 1 and 0 <= n - 2^k - k < 2^k, = a(n-2^k) if k >= 1 and n - 2^k - k = -1, or = 0 if n = 0 (and exactly one of the three conditions is true for any n >= 0).

"Equivalently, a(2^k + k + x) = 2^k + a(k+x) if 0 <= x < 2^k, = a(k+x) if x = -1 (for each n >= 0, there is a unique k, x such that 2^k + k + x = n, k >= 0, -1 <= x < 2^k). This recurrence follows immediately from the definition.

"The recurrence captures three observed facts about a: a(2^k + k - 1) = a(k-1); a consists of blocks of length 2^k of A105025 interspersed with terms of a; a(n) = n - Sum_{ k >= 0, 2^{k+1} <= n, n = k mod 2^(k+1) } 2^(k+1)."

Start with the binary numbers:

........0

........1

.......10

.......11

......100

......101

......110

......111

.....1000

.........

and read downwards to the right, getting 0, 1, 0, 11, 10, 1, 100, 111, ...

f:= proc (n) local t1, l; t1 := n; for l from 0 to n do if `mod`(n-l, 2^(l+1)) = 0 and n >= 2^(l+1) then t1 := t1-2^(l+1) fi; od; t1; end proc;

f[n_] := Block[{k = 0, s = 0}, While[2^(k + 1) < n + 1, If[ Mod[n, 2^(k + 1)] == k, s = s + 2^(k + 1)]; k++ ]; n - s]; Table[ f[n], {n, 0, 75}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 06 2005)

Analogue of A102370. Cf. A105034, A105025.

Cf. triangular array in A103589.

Sequence in context: A127671 A104509 A117212 this_sequence A092486 A159966 A119263

Adjacent sequences: A105030 A105031 A105032 this_sequence A105034 A105035 A105036

nonn

N. J. A. Sloane (njas(AT)research.att.com), Apr 04 2005