A137821 Numbers k such that Sum_{j=1..2k} Catalan(j) == 0 (mod 3).

1, 4, 6, 13, 15, 18, 19, 40, 42, 45, 46, 54, 55, 58, 60, 121, 123, 126, 127, 135, 136, 139, 141, 162, 163, 166, 168, 175, 177, 180, 181, 364, 366, 369, 370, 378, 379, 382, 384, 405, 406, 409, 411, 418, 420, 423, 424, 486, 487, 490, 492, 499, 501, 504, 505

Offset

1,2

Comments

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.

Links

M. F. Hasler, Table of n, a(n) for n = 1..499.

Formula

a(n) = A107755(n)/2 = Sum_{k=0..n} A137822(k).

a(2^j) = 2 a(2^j-1) + 1 (resp. +2) for j even (resp. odd).

Sum_{k=1..2n} Catalan(k) = Sum_{k=1..n} Catalan(2k-1) * (10k-1)/(2k+1), thus:

{ a(m) } = { n>0 | Sum_{k=1..n} Catalan(2k-1) * (10k-1)/(2k+1) == 0 (mod 3) }.

Prog

(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})

Crossrefs

Cf. A107755 (twice this), A137822-A137824.

Sequence in context: A354089 A247787 A074165 * A010061 A280557 A266665

Adjacent sequences: A137818 A137819 A137820 * A137822 A137823 A137824

Keyword

nonn

Author

M. F. Hasler, Feb 25 2008

Status

approved