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

Author

N. J. A. Sloane, Jun 11 2005

Keywords

• nonn
• easy

Links

• R. J. Mathar, Table of n, a(n) for n = 1..319.
• Y. More, Problem 11165, Amer. Math. Monthly, 112 (2005), 568.

Crossrefs

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

Programs

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 *)

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

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

Extensions

More terms from Emeric Deutsch, Jun 12 2005

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