A111018 Indices of Catalan numbers that are divisible by 3.

5, 6, 7, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 32, 33, 34, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 86, 87, 88, 95, 96, 97, 98, 99, 100

Offset: 1

Author

Robert G. Wilson v, Sep 09 2005

Keywords

• nonn

Comments

Conjecture: The sequence contains all numbers n such that n and n+1 are both in the sequence A074940, or, equivalently, such that neither n nor n+1 is in the sequence A005836. - L. Edson Jeffery, Dec 02 2015

The asymptotic density of this sequence is 1 (Burns, 2016). - Amiram Eldar, Jan 26 2021

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Rob Burns, Asymptotic density of Catalan numbers modulo 3 and powers of 2, arXiv:1611.03705 [math.NT], 2016.

Crossrefs

Cf. A000108.

Cf. A085296, A265100, A265104.

Programs

Magma

[n: n in [1..200] | Binomial(2*n, n) div (n+1) mod 3 eq 0]; // Vincenzo Librandi, Dec 03 2015

Mathematica

Flatten[Position[CatalanNumber[Range[100]],?(Divisible[#,3]&)]] (* _Harvey P. Dale, Oct 25 2011 *)
Select[Range@100, Divisible[CatalanNumber@#,3]&] (* Vladimir Reshetnikov, Nov 06 2015 *)

PARI

for(n=0, 1e3, if(binomial(2*n, n)/(n+1) % 3 == 0, print1(n, ", "))) \\ Altug Alkan, Dec 02 2015