A289682 Catalan numbers read modulo 16.

1, 1, 2, 5, 14, 10, 4, 13, 6, 14, 12, 2, 12, 4, 8, 13, 6, 6, 12, 6, 4, 12, 8, 2, 12, 12, 8, 4, 8, 8, 0, 13, 6, 6, 12, 14, 4, 12, 8, 6, 4, 4, 8, 12, 8, 8, 0, 2, 12, 12, 8, 12, 8, 8, 0, 4, 8, 8, 0, 8, 0, 0, 0, 13, 6, 6, 12, 14, 4, 12, 8, 14, 4, 4, 8, 12, 8, 8, 0, 6, 4, 4, 8, 4, 8, 8, 0, 12

Offset

0,3

Comments

Conjecture: a(2^n-1) = 13 and a(2^n) = 6 for n >= 3. - Robert Israel, Jul 09 2017

Links

Robert Israel, 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.

Shu-Chung Liu and Jean C.-C. Yeh, Catalan numbers modulo 2^k, J. Int. Seq., Vol. 13 (2010), Article 10.5.4, Theorem 5.5.

Formula

a(n) = A000108(n) mod 16.

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 0 (Burns, 2016). - Amiram Eldar, Jan 26 2021

Maple

seq ( modp(A000108(n), 16), n=0..120) ;

Mathematica

Table[Mod[CatalanNumber[n], 16], {n, 0, 100}] (* Vincenzo Librandi, Jul 10 2017 *)

Prog

(PARI) a(n) = (binomial(2*n, n)/(n+1)) % 16; \\ Michel Marcus, Jul 09 2017
(Magma) [Catalan(n) mod 16: n in [0..100]]; // Vincenzo Librandi, Jul 10 2017

Crossrefs

Cf. A000108, A036987 (mod 2), A073267 (mod 4), A159987 (mod 8).

Cf. A048881 (2-adic valuation of A000108).

Sequence in context: A279958 A348881 A324982 * A151854 A146526 A120626

Adjacent sequences: A289679 A289680 A289681 * A289683 A289684 A289685

Keyword

nonn,easy

Author

R. J. Mathar, Jul 09 2017

Status

approved