A301453 a(n) is the number of ways of writing the binary expansion of n as a concatenation of nonempty substrings with no two consecutive equal substrings.

1, 1, 2, 1, 3, 4, 3, 3, 6, 7, 7, 6, 5, 6, 6, 4, 10, 13, 14, 11, 11, 14, 14, 12, 9, 11, 11, 9, 9, 12, 10, 7, 17, 23, 26, 20, 20, 26, 25, 21, 23, 26, 28, 22, 22, 27, 26, 20, 16, 20, 22, 17, 17, 22, 20, 18, 18, 21, 23, 18, 16, 20, 17, 14, 31, 40, 46, 36, 39, 49

Offset: 0

Author

Rémy Sigrist, Apr 08 2018

Keywords

• nonn
• base

Comments

Leading zeros in the binary expansion of n are ignored.

The value a(0) = 1 corresponds to the empty concatenation.

The following sequences f correspond to the numbers of ways of writing the binary expansion of a number as a concatenation of substrings with some specific features:

f f(2^n-1) Features

------- -------- --------

A215244 A011782 Substrings are palindromes.

A301453 A003242 This sequence; no two consecutive equal substrings.

A302395 A032020 All substrings are distinct.

A302436 A000012 Substrings with Hamming weight at most 1.

A302437 A000045 Substrings with Hamming weight at most 2.

A302439 A000012 Substrings are aperiodic.

For any such sequence f, the function n -> f(2^n-1) corresponds to a composition of n.

Examples

			For n = 19: the binary expansion of 19, "10011", can be split in 11 ways into nonempty substrings with no two consecutive equal substrings:
- (10011),
- (1001)(1),
- (100)(11),
- (10)(011),
- (10)(01)(1),
- (10)(0)(11),
- (1)(0011),
- (1)(001)(1),
- (1)(00)(11),
- (1)(0)(011),
- (1)(0)(01)(1).
Hence a(19) = 11.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, Scatterplot of the second ordinal transform of the first 1000000 terms
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000012, A000045, A003242, A011782, A032020, A215244, A301453, A302395, A302436, A302437, A302439.

Programs

PARI

a(n{, pp=0}) = if (n==0, return (1), my (v=0, p=1); while (n, p=(p*2) + (n%2); n\=2; if (p!=pp, v+=a(n, p))); return (v))

Formula

a(2^n - 1) = A003242(n) for any n >= 0.