%I #18 Dec 13 2023 08:44:33
%S 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,
%T 30,31,33,34,35,37,38,39,41,42,43,45,46,47,49,50,52,53,54,56,57,58,60,
%U 61,62,64,65,66,68,69,71,72,73,75,76,77,79,80
%N a(n) = floor(log(Catalan(n))).
%C Largest integer m such that e^m < C(n), the n-th Catalan number, where e = exp(1) is the Euler number.
%H Stanislav Sykora, <a href="/A260816/b260816.txt">Table of n, a(n) for n = 0..2000</a>
%F a(n) = floor(log(C(n))).
%F 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
%e a(5) = 3 because e^3 < C(3) = 42 < e^4.
%p seq(floor(log(binomial(2*n,n)/(n+1))), n=0 .. 100); # _Robert Israel_, Aug 19 2015
%t f[n_] := Floor@ Log@ CatalanNumber@ n; Array[f, 70, 0] (* _Robert G. Wilson v_, Aug 18 2015 *)
%o (PARI) a(n)=floor(log(binomial(2*n,n)/(n+1)))
%o \\ Use realprecision > number of digits of C(max n)
%o (Magma) [Floor(Log(Binomial(2*n, n)/(n+1))): n in [0.. 65]]; // _Vincenzo Librandi_, Aug 20 2015
%Y Cf. A000108, A001113.
%K nonn,less
%O 0,5
%A _Stanislav Sykora_, Jul 31 2015