A107755 Numbers k such that Sum_{j=1..k} Catalan(j) == 0 (mod 3).

2, 8, 12, 26, 30, 36, 38, 80, 84, 90, 92, 108, 110, 116, 120, 242, 246, 252, 254, 270, 272, 278, 282, 324, 326, 332, 336, 350, 354, 360, 362, 728, 732, 738, 740, 756, 758, 764, 768, 810, 812, 818, 822, 836, 840, 846, 848, 972, 974, 980, 984, 998, 1002, 1008, 1010

Offset

1,1

Links

R. J. Mathar, Table of n, a(n) for n = 1..319.

Y. More, Problem 11165, Amer. Math. Monthly, 112 (2005), 568.

Formula

a(2^j) = 2*a(2^j-1) + 2 (resp. + 4) if j is even (resp. odd). - M. F. Hasler, Feb 25 2008

a(n) = 2*Sum_{i=1..n} A137822(i). - M. F. Hasler, Mar 16 2008

{n: A137993(n-1) = 0}. - R. J. Mathar, Jul 07 2009

Maple

A107755 := proc(n) option remember ; local a; if n = 1 then 2; else for a from A107755(n-1)+1 do if add(A000108(k), k=1..a) mod 3 = 0 then RETURN(a) ; fi ; od: fi ; end: # R. J. Mathar, Feb 25 2008
c:=n->binomial(2*n, n)/(n+1): s:=0: for n from 1 to 1500 do s:=s+c(n): a[n]:=s mod 3: od: A:=[seq(a[n], n=1..1500)]: p:=proc(n) if A[n]=0 then n else fi end: seq(p(n), n=1..1500); # Emeric Deutsch, Jun 12 2005

Mathematica

s0 = s2 = {}; s = 0; Do[s = Mod[s + (2 n)!/n!/(n + 1)!, 3]; Switch[ Mod[s, 3], 0, AppendTo[s0, n], 2, AppendTo[s2, n]], {n, 1055}]; s0 (* Robert G. Wilson v, Jun 14 2005 *)
Flatten[Position[Accumulate[CatalanNumber[Range[1100]]], _?(Divisible[ #, 3]&)]] (* Harvey P. Dale, Feb 07 2016 *)

Prog

(PARI) n=0; s=Mod(0, 3); A107755=vector(100, i, if( bitand(i, i-1), while(n++ && s+=binomial(2*n, n)/(n+1), ), s=Mod(0, 3); n=2*n+2+(log(i+.5)\log(2)%2)*2 ); /*print1(n", "); */ n) \\ M. F. Hasler, Feb 25 2008
(PARI) A107755(n)=sum( i=1, n, A137822(i) )*2 /* allows computation of a(10^4) in one second */ \\ M. F. Hasler, Mar 16 2008

Crossrefs

Cf. A000108, A107756, A107757, A108784, A137821, A137822, A137823, A137824.

Sequence in context: A108978 A269968 A135957 * A268836 A027718 A115102

Adjacent sequences: A107752 A107753 A107754 * A107756 A107757 A107758

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 11 2005

Extensions

More terms from Emeric Deutsch, Jun 12 2005

Corrected & extended by M. F. Hasler and R. J. Mathar, Feb 25 2008

Status

approved