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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 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.)