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 A039969 An example of a d-perfect sequence: a(n) = Catalan(n) mod 3. 11
 1, 1, 2, 2, 2, 0, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS This is A006996 with all its terms repeated three times, except the initial term only twice. A006996 is a fixed point of the morphism 0 -> 000, 1 -> 120, 2 -> 210. [The original comment edited by Antti Karttunen, Aug 14 2017] Equals Catalan(n) mod 3. (Cf. A000108.) - Paul D. Hanna, Jun 20 2003 [confirmed by Christian G. Bower, Jun 12 2005] Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). LINKS Antti Karttunen, Table of n, a(n) for n = 0..10000 Rob Burns, Asymptotic density of Catalan numbers modulo 3 and powers of 2, arXiv:1611.03705 [math.NT], 2016. David Kohel, San Ling and Chaoping Xing, Explicit Sequence Expansions, in: C. Ding, T. Helleseth and H. Niederreiter (eds.), Sequences and their Applications, Proceedings of SETA'98 (Singapore, 1998), Discrete Mathematics and Theoretical Computer Science, 1999, pp. 308-317; alternative link. Index entries for sequences that are fixed points of mappings FORMULA a(n) = ((-1)^(n+1)*A001006(n-1)) mod 3, for n>0. - Christian G. Bower, Jun 12 2005 a(n) = a(n-1) if n == 0 or 1 (mod 3). a(n) = 0 if n == 5,6, or 7 (mod 9). - Robert Israel, Sep 20 2015 a(3n) = A006996(n). - Antti Karttunen, Aug 14 2017 Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 0 (Burns, 2016). - Amiram Eldar, Jan 26 2021 MAPLE seq(binomial(2*n, n)/(n+1) mod 3, n = 0 .. 100); # Robert Israel, Sep 20 2015 MATHEMATICA Take[ Flatten[ Nest[ Flatten[ # /. {1 -> {1, 2, 0}, 2 -> {2, 1, 0}, 0 -> {0, 0, 0}}] &, {1}, 4] /. {1 -> {1, 1, 1}, 2 -> {2, 2, 2}, 0 -> {0, 0, 0}}], {2, 106}] (* or *) Table[ Mod[ Binomial[ 2n, n]/(n + 1), 3], {n, 0, 104}] (* Robert G. Wilson v, Sep 09 2005 *) Mod[CatalanNumber[Range[0, 110]], 3] (* Harvey P. Dale, Oct 23 2017 *) PROG (Magma) [Catalan(n) mod 3: n in [1..80]]; // Vincenzo Librandi, Jul 14 2015 (PARI) A039969(n) = ((binomial(2*n, n)/(n+1))%3); \\ Antti Karttunen, Aug 13 2017 CROSSREFS Cf. A000108, A001006, A010872, A039972, A067397. Cf. A006996 (trisection). Sequence in context: A028953 A348648 A037865 * A039967 A258133 A327152 Adjacent sequences: A039966 A039967 A039968 * A039970 A039971 A039972 KEYWORD nonn AUTHOR N. J. A. Sloane EXTENSIONS More terms from Christian G. Bower, Jun 12 2005 Offset corrected from 1 to 0 by Antti Karttunen, Aug 13 2017 STATUS approved

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Last modified August 5 19:08 EDT 2024. Contains 374954 sequences. (Running on oeis4.)