cp's OEIS Frontend

A159981 Catalan numbers read modulo 4.

%I A159981 #25 Jan 26 2021 04:00:52
%S A159981 1,1,2,1,2,2,0,1,2,2,0,2,0,0,0,1,2,2,0,2,0,0,0,2,0,0,0,0,0,0,0,1,2,2,
%T A159981 0,2,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,0,2,
%U A159981 0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0
%N A159981 Catalan numbers read modulo 4.
%C A159981 Essentially the same as A073267. - _R. J. Mathar_, May 25 2009
%H A159981 Antti Karttunen, <a href="/A159981/b159981.txt">Table of n, a(n) for n = 0..10000</a>
%H A159981 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 A159981 Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh, <a href="https://doi.org/10.1016/j.ejc.2007.06.019">Catalan and Motzkin numbers modulo 4 and 8</a>, European Journal of Combinatorics, Vol. 29, No. 6 (2008), pp. 1449-1466. [From _R. J. Mathar_, Apr 30 2009]
%H A159981 Eric Rowland and Reem Yassawi, <a href="https://doi.org/10.1016/j.aam.2016.11.005">Profinite automata</a>, Advances in Applied Mathematics, Vol. 85 (2017), pp. 60-83; <a href="https://arxiv.org/abs/1403.7659">arXiv preprint</a>, arXiv:1403.7659 [math.DS], 2014-2016. See p. 4, 6 and 7.
%F A159981 a(n) = A000108(n) mod 4.
%F A159981 Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 0 (Burns, 2016). - _Amiram Eldar_, Jan 26 2021
%p A159981 A000108 := proc(n) binomial(2*n,n)/(n+1) ; end: A159981 := proc(n) A000108(n) mod 4 ; end: seq(A159981(n),n=0..120) ; # _R. J. Mathar_, Apr 30 2009
%t A159981 Mod[CatalanNumber[Range[0,110]],4] (* _Harvey P. Dale_, Oct 05 2011 *)
%o A159981 (PARI) A159981(n) = (binomial(2*n, n)/(n+1))%4; \\ _Antti Karttunen_, Jan 17 2017
%Y A159981 Cf. A000108, A073267.
%K A159981 easy,nonn
%O A159981 0,3
%A A159981 _Philippe Deléham_, Apr 28 2009
%E A159981 Extended by _R. J. Mathar_, Apr 30 2009