cp's OEIS Frontend

A289682 Catalan numbers read modulo 16.

%I A289682 #23 Sep 08 2022 08:46:19
%S A289682 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,
%T A289682 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,
%U A289682 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
%N A289682 Catalan numbers read modulo 16.
%C A289682 Conjecture: a(2^n-1) = 13 and a(2^n) = 6 for n >= 3. - _Robert Israel_, Jul 09 2017
%H A289682 Robert Israel, <a href="/A289682/b289682.txt">Table of n, a(n) for n = 0..10000</a>
%H A289682 Rob Burns, <a href="https://arxiv.org/abs/1611.03705">Asymptotic density of Catalan numbers modulo 3 and powers of 2</a>, arXiv:1611.03705 [math.NT], 2016.
%H A289682 Shu-Chung Liu and Jean C.-C. Yeh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Liu2/liu6.html">Catalan numbers modulo 2^k</a>, J. Int. Seq., Vol. 13 (2010), Article 10.5.4, Theorem 5.5.
%F A289682 a(n) = A000108(n) mod 16.
%F A289682 Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 0 (Burns, 2016). - _Amiram Eldar_, Jan 26 2021
%p A289682 seq ( modp(A000108(n),16),n=0..120) ;
%t A289682 Table[Mod[CatalanNumber[n], 16], {n, 0, 100}] (* _Vincenzo Librandi_, Jul 10 2017 *)
%o A289682 (PARI) a(n) = (binomial(2*n, n)/(n+1)) % 16; \\ _Michel Marcus_, Jul 09 2017
%o A289682 (Magma) [Catalan(n) mod 16: n in [0..100]]; // _Vincenzo Librandi_, Jul 10 2017
%Y A289682 Cf. A000108, A036987 (mod 2), A073267 (mod 4), A159987 (mod 8).
%Y A289682 Cf. A048881 (2-adic valuation of A000108).
%K A289682 nonn,easy
%O A289682 0,3
%A A289682 _R. J. Mathar_, Jul 09 2017