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

Author

R. J. Mathar, Jul 09 2017

Keywords

• nonn
• easy

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.

Crossrefs

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

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

Programs

Magma

[Catalan(n) mod 16: n in [0..100]]; // Vincenzo Librandi, Jul 10 2017

Maple

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

Mathematica

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

PARI

a(n) = (binomial(2*n, n)/(n+1)) % 16; \\ Michel Marcus, Jul 09 2017

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