A159987 Catalan numbers read modulo 8.
1, 1, 2, 5, 6, 2, 4, 5, 6, 6, 4, 2, 4, 4, 0, 5, 6, 6, 4, 6, 4, 4, 0, 2, 4, 4, 0, 4, 0, 0, 0, 5, 6, 6, 4, 6, 4, 4, 0, 6, 4, 4, 0, 4, 0, 0, 0, 2, 4, 4, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 5, 6, 6, 4, 6, 4, 4, 0, 6, 4, 4, 0, 4, 0, 0, 0, 6, 4, 4, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 4, 4, 0, 4, 0, 0, 0, 4, 0
Offset: 0
Links
- 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.
Programs
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Maple
A000108 := proc(n) binomial(2*n,n)/(n+1) ; end: A159987 := proc(n) A000108(n) mod 8 ; end: seq(A159987(n),n=0..120) ; # R. J. Mathar, Apr 30 2009
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Mathematica
Table[Mod[CatalanNumber[n], 8], {n, 0, 100}] (* Amiram Eldar, Jan 26 2021 *)
Formula
a(n) = A000108(n) mod 8.
a(n) == A159981(n) (mod 4). - R. J. Mathar, Apr 30 2009
Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 0 (Burns, 2016). - Amiram Eldar, Jan 26 2021