A355340 a(0) = 0; for n >= 1, a(n) = a(n-1) XOR A001511(n), where XOR denotes bitwise exclusive-or (A003987) and A001511 is the binary ruler function.

0, 1, 3, 2, 1, 0, 2, 3, 7, 6, 4, 5, 6, 7, 5, 4, 1, 0, 2, 3, 0, 1, 3, 2, 6, 7, 5, 4, 7, 6, 4, 5, 3, 2, 0, 1, 2, 3, 1, 0, 4, 5, 7, 6, 5, 4, 6, 7, 2, 3, 1, 0, 3, 2, 0, 1, 5, 4, 6, 7, 4, 5, 7, 6, 1, 0, 2, 3, 0, 1, 3, 2, 6, 7, 5, 4, 7, 6, 4, 5, 0, 1, 3, 2, 1, 0, 2, 3, 7, 6, 4, 5, 6, 7, 5, 4, 2, 3, 1, 0, 3, 2, 0, 1, 5

Offset: 0

Author

Peter Munn, Jun 29 2022

Keywords

• nonn
• base
• easy

Comments

Related to the Thue-Morse sequence, A010060, which gives the rightmost binary bit of each term. The next bit is given by the closely related A269723.

If we replace A001511(n) in the definition by A006519(n) = 2^(A001511(n)-1) we get Gray code (A003188).

Interesting symmetries of the sequence seem more apparent with the terms aligned in suitable periods, such as the arrangement in the example section.

Examples

			Initial terms arranged in periods of 16, with deliberate periodic spacing:
  0,1,3,2,  1,0,2,3,     7,6,4,5,  6,7,5,4,
  1,0,2,3,  0,1,3,2,     6,7,5,4,  7,6,4,5,
  3,2,0,1,  2,3,1,0,     4,5,7,6,  5,4,6,7,
  2,3,1,0,  3,2,0,1,     5,4,6,7,  4,5,7,6,
.
  1,0,2,3,  0,1,3,2,     6,7,5,4,  7,6,4,5,
  0,1,3,2,  1,0,2,3,     7,6,4,5,  6,7,5,4,
  2,3,1,0,  3,2,0,1,     5,4,6,7,  4,5,7,6,
  3,2,0,1,  2,3,1,0,     4,5,7,6,  5,4,6,7,
...
Note that when the arrangement is partitioned regularly into 2 X 2, 4 X 4 or 8 X 8 squares, the terms on any diagonal of a square share the same value. Note also the symmetry of the terms on the squares' circumferences.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..16383
• Rémy Sigrist, Colored representation of the first 2^15 terms as 128 rows of 256 terms (the color is function of a(x + 256*y), x = 0..255, y = 0..127)
• Index entries for sequences related to binary expansion of n

Crossrefs

Comparable sequences: A010060, A261283, A269723.

Cf. A001511, A003188, A003987, A006519.

Positions of: odd numbers: A000069, even numbers: A001969, previously unseen numbers: A253317 (apparently).

Programs

Mathematica

Block[{k = 0}, NestList[BitXor[#, IntegerExponent[k += 2, 2]] &, 0, 100]] (* Paolo Xausa, May 29 2024 *)

Formula

A010060(n) = a(n) mod 2.

A269723(n) = floor(a(n)/2) mod 2.