A302437 -id:A302437 - cp's OEIS frontend

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

Rémy Sigrist, Apr 08 2018

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.

			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.

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

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

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))

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

cp's OEIS Frontend

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.

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