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 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 (list; graph; refs; listen; history; text; internal format)
 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: 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

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Last modified May 18 15:25 EDT 2021. Contains 343995 sequences. (Running on oeis4.)