A331855 a(n) is the number of distinct values obtained by partitioning the binary representation of n into consecutive blocks, and then reversing those blocks.

1, 1, 2, 1, 3, 3, 3, 1, 4, 6, 5, 4, 5, 4, 4, 1, 5, 10, 9, 9, 8, 8, 9, 5, 7, 9, 8, 5, 7, 5, 5, 1, 6, 15, 14, 16, 12, 16, 18, 12, 11, 16, 13, 12, 15, 13, 14, 6, 9, 16, 15, 13, 13, 12, 12, 6, 10, 12, 11, 6, 9, 6, 6, 1, 7, 21, 20, 25, 18, 27, 30, 22, 16, 27, 25

Offset: 0

Author

Rémy Sigrist, Jan 29 2020

Keywords

• nonn
• base

Examples

			For n = 6:
- the binary representation of 6 is "110",
- we can split it in 4 ways:
      "110" -> "011" -> 3
      "1" and "10" -> "1" and "01" -> 5
      "11" and "0" -> "11" and "0" -> 6
      "1" and "1" and "0" -> "1" and "1" and "0" -> 6
- we have 3 distinct values,
- hence a(6) = 3.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..16384
• Rémy Sigrist, PARI program for A331855
• Index entries for sequences related to binary expansion of n

Crossrefs

See A331851 for similar sequences.

See A331856 and A331857 for the least and greatest values, respectively.

Cf. A000096, A000217.

Programs

PARI

See Links section.

Formula

a(2^k-1) = 1 for any k >= 0.

a(2^k) = k+1 for any k >= 0.

a(2^k+1) = A000217(k) for any k > 0.

a(2^k+2) = A000096(k-1) for any k > 3.

a(2^k+3) = (k-1)^2 for any k > 1.