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

0,3

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.

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

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 *)

Prog

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

Crossrefs

Cf. A000108, A073267.

Sequence in context: A175609 A038717 A073267 * A071858 A122864 A140084

Adjacent sequences: A159978 A159979 A159980 * A159982 A159983 A159984

Keyword

easy,nonn

Author

Philippe Deléham, Apr 28 2009

Extensions

Extended by R. J. Mathar, Apr 30 2009

Status

approved