A224446 - OEIS

%I #15 Apr 07 2014 03:51:20

%S 1,4,64,512,16384,131072,2097152,16777216,1073741824,8589934592,

%T 137438953472,1099511627776,35184372088832,281474976710656,

%U 4503599627370496,36028797018963968,4611686018427387904,36893488147419103232

%N Denominators of certain rationals approximating sqrt(3).

%C See the numerator sequence A224445. The rationals r(n) are the partial sums of the series 2*sqrt(1 - 1/4), representing sqrt(3).

%C Looks as if a(n) are all powers of 2, a(n) = 2^b(n) with b(n) = 0, 2, 6, 9, 14, 17, 21, 24, 30, 33, 37, 40, 45, 48, ... - _Peter Luschny_, Apr 05 2014

%D H. K. Strick, Geschichten aus der Mathematik, Spektrum Spezial 2/2009, p. 45 (on Newton).

%H Vincenzo Librandi, <a href="/A224446/b224446.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = denominator(r(n)), n >= 0, with the rationals (in lowest terms)  r(n) = 2*(1 - 2*sum(C(k-1)/2^(4*k)),k=1..n), with the Catalan numbers C(n) = A000108(n).

%e a(2) = 64 because r(2) =  111/64 = A224445(2)/a(2).

%p A224446 := proc(n) (x/(exp(x)-1))^(3/2)*exp(x/2);

%p -pochhammer(1/2,n-1)*coeff(series(%,x,n+2),x,n); denom(%) end:

%p seq(A224446(n),n=0..17); # _Peter Luschny_, Apr 05 2014

%t r[n_] := 2*(1 - 2*Sum[ CatalanNumber[k - 1]/4^(2*k), {k, 1, n}]); Table[r[n], {n, 0, 17}] // Denominator (* _Jean-François Alcover_, Apr 09 2013 *)

%Y Cf. A224445, A000108, A002194.

%K nonn,frac

%O 0,2

%A _Wolfdieter Lang_, Apr 09 2013