A107755 - OEIS

%I #33 Feb 10 2021 01:16:43

%S 2,8,12,26,30,36,38,80,84,90,92,108,110,116,120,242,246,252,254,270,

%T 272,278,282,324,326,332,336,350,354,360,362,728,732,738,740,756,758,

%U 764,768,810,812,818,822,836,840,846,848,972,974,980,984,998,1002,1008,1010

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

%H R. J. Mathar, <a href="/A107755/b107755.txt">Table of n, a(n) for n = 1..319</a>.

%H Y. More, <a href="http://www.jstor.org/stable/30037533">Problem 11165</a>, Amer. Math. Monthly, 112 (2005), 568.

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

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

%F {n: A137993(n-1) = 0}. - _R. J. Mathar_, Jul 07 2009

%p 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

%p 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

%t 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 *)

%t Flatten[Position[Accumulate[CatalanNumber[Range[1100]]],_?(Divisible[ #,3]&)]] (* _Harvey P. Dale_, Feb 07 2016 *)

%o (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

%o (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

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

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_, Jun 11 2005

%E More terms from _Emeric Deutsch_, Jun 12 2005

%E Corrected & extended by _M. F. Hasler_ and _R. J. Mathar_, Feb 25 2008