A159981 Catalan numbers read modulo 4.

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, 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, 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

Offset: 0

Author

Philippe Deléham, Apr 28 2009

Keywords

• easy
• nonn

Comments

Essentially the same as A073267. - R. J. Mathar, May 25 2009

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.
• Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh, Catalan and Motzkin numbers modulo 4 and 8, European Journal of Combinatorics, Vol. 29, No. 6 (2008), pp. 1449-1466. [From _R. J. Mathar_, Apr 30 2009]
• Eric Rowland and Reem Yassawi, Profinite automata, Advances in Applied Mathematics, Vol. 85 (2017), pp. 60-83; arXiv preprint, arXiv:1403.7659 [math.DS], 2014-2016. See p. 4, 6 and 7.

Crossrefs

Cf. A000108, A073267.

Programs

Maple

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

Mathematica

Mod[CatalanNumber[Range[0,110]],4] (* Harvey P. Dale, Oct 05 2011 *)

PARI

A159981(n) = (binomial(2*n, n)/(n+1))%4; \\ Antti Karttunen, Jan 17 2017

Formula

a(n) = A000108(n) mod 4.

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

Extensions

Extended by R. J. Mathar, Apr 30 2009