A260816 a(n) = floor(log(Catalan(n))).

0, 0, 0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 64, 65, 66, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80

Offset

0,5

Comments

Largest integer m such that e^m < C(n), the n-th Catalan number, where e = exp(1) is the Euler number.

Links

Stanislav Sykora, Table of n, a(n) for n = 0..2000

Formula

a(n) = floor(log(C(n))).

For n >= 1, a(n) is either floor(2*log(2)*n - (3/2)*log(n)) or floor(2*log(2)*n - (3/2)*log(n)) - 1. - Robert Israel, Aug 19 2015

Example

a(5) = 3 because e^3 < C(3) = 42 < e^4.

Maple

seq(floor(log(binomial(2*n, n)/(n+1))), n=0 .. 100); # Robert Israel, Aug 19 2015

Mathematica

f[n_] := Floor@ Log@ CatalanNumber@ n; Array[f, 70, 0] (* Robert G. Wilson v, Aug 18 2015 *)

Prog

(PARI) a(n)=floor(log(binomial(2*n, n)/(n+1)))
\\ Use realprecision > number of digits of C(max n)
(Magma) [Floor(Log(Binomial(2*n, n)/(n+1))): n in [0.. 65]]; // Vincenzo Librandi, Aug 20 2015

Crossrefs

Cf. A000108, A001113.

Sequence in context: A039160 A175439 A048264 * A285598 A275804 A141825

Adjacent sequences: A260813 A260814 A260815 * A260817 A260818 A260819

Keyword

nonn,less

Author

Stanislav Sykora, Jul 31 2015

Status

approved