

A137821


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


7



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
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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^j1) + 1 (resp. +2) for j even (resp. odd).
Sum_{k=1..2n} Catalan(k) = Sum_{k=1..n} Catalan(2k1) * (10k1)/(2k+1), thus:
{ a(m) } = { n>0  Sum_{k=1..n} Catalan(2k1) * (10k1)/(2k+1) == 0 (mod 3) }.


PROG

(PARI) n=0; A137821=vector(499, i, { if( bitand(i, i1), while(n++ & s+=binomial(4*n2, 2*n1)/(2*n)*(10*n1)/(2*n+1), ), s=Mod(0, 3); n=2*n+1+log(i+.5)\log(2)%2 ); n})


CROSSREFS

Cf. A107755 (twice this), A137822A137824.
Sequence in context: A191199 A247787 A074165 * A010061 A280557 A266665
Adjacent sequences: A137818 A137819 A137820 * A137822 A137823 A137824


KEYWORD

nonn


AUTHOR

M. F. Hasler, Feb 25 2008


STATUS

approved



