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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. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

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

FORMULA

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

EXAMPLE

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.

PROG

(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))

CROSSREFS

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

Sequence in context: A237124 A233547 A122530 * A278340 A324749 A022466

Adjacent sequences:  A301450 A301451 A301452 * A301454 A301455 A301456

KEYWORD

nonn,base

AUTHOR

Rémy Sigrist, Apr 08 2018

STATUS

approved

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Last modified January 24 01:05 EST 2020. Contains 331178 sequences. (Running on oeis4.)