A302436 a(n) is the number of ways of writing the binary expansion of n as a concatenation of nonempty substrings with Hamming weight at most 1.

1, 1, 2, 1, 4, 3, 2, 1, 8, 7, 6, 3, 4, 3, 2, 1, 16, 15, 14, 7, 12, 9, 6, 3, 8, 7, 6, 3, 4, 3, 2, 1, 32, 31, 30, 15, 28, 21, 14, 7, 24, 21, 18, 9, 12, 9, 6, 3, 16, 15, 14, 7, 12, 9, 6, 3, 8, 7, 6, 3, 4, 3, 2, 1, 64, 63, 62, 31, 60, 45, 30, 15, 56, 49, 42, 21

Offset

0,3

Comments

Leading zeros in the binary expansion of n are ignored.

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

See A301453 for similar sequences.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Formula

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

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

a(2*n) = 2*a(n) for any n > 0.

Example

For n = 9: the binary expansion of 9, "1001", can be split in 7 ways into nonempty substrings with Hamming weight at most 1:
- (100)(1),
- (10)(01),
- (10)(0)(1),
- (1)(001),
- (1)(00)(1),
- (1)(0)(01),
- (1)(0)(0)(1).
Hence a(9) = 7.

Prog

(PARI) a(n) = if (n==0, return (1), my (v=0, h=0); while (n, h += n%2; n\=2; if (h>1, break, v+=a(n))); return (v))

Crossrefs

Cf. A301453.

Sequence in context: A253515 A319521 A281589 * A283167 A355807 A355808

Adjacent sequences: A302433 A302434 A302435 * A302437 A302438 A302439

Keyword

nonn,base

Author

Rémy Sigrist, Apr 08 2018

Status

approved