%I #8 Jan 05 2020 22:22:13
%S 1,4,6,13,15,18,19,40,42,45,46,54,55,58,60,121,123,126,127,135,136,
%T 139,141,162,163,166,168,175,177,180,181,364,366,369,370,378,379,382,
%U 384,405,406,409,411,418,420,423,424,486,487,490,492,499,501,504,505
%N Numbers k such that Sum_{j=1..2k} Catalan(j) == 0 (mod 3).
%C It would be natural to prepend an initial term a(1)=0 (for which the sum is to be considered empty, thus zero), but we omit it to avoid confusion w.r.t. indices of A107755.
%H M. F. Hasler, <a href="/A137821/b137821.txt">Table of n, a(n) for n = 1..499</a>.
%F a(n) = A107755(n)/2 = Sum_{k=0..n} A137822(k).
%F a(2^j) = 2 a(2^j-1) + 1 (resp. +2) for j even (resp. odd).
%F Sum_{k=1..2n} Catalan(k) = Sum_{k=1..n} Catalan(2k-1) * (10k-1)/(2k+1), thus:
%F { a(m) } = { n>0 | Sum_{k=1..n} Catalan(2k-1) * (10k-1)/(2k+1) == 0 (mod 3) }.
%o (PARI) n=0; A137821=vector(499,i,{ if( bitand(i,i-1), while(n++ & s+=binomial(4*n-2,2*n-1)/(2*n)*(10*n-1)/(2*n+1),),s=Mod(0,3); n=2*n+1+log(i+.5)\log(2)%2 ); n})
%Y Cf. A107755 (twice this), A137822-A137824.
%K nonn
%O 1,2
%A _M. F. Hasler_, Feb 25 2008