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%I #10 Jul 19 2013 13:23:17
%S 2,7,111,887,28379,227025,3632379,29058999,1859775507,14878203341,
%T 238051251025,1904410004001,60941120098639,487528960737109,
%U 7800463371608019,62403706972529847,7987674492474125571,63901395939775325733
%N Numerators of certain rationals approximating sqrt(3).
%C The corresponding denominators are given in A224446.
%C The rationals r(n) are the partial sums of the series 2*sqrt(1 - 1/4) which represents sqrt(3).
%D H. K. Strick, Geschichten aus der Mathematik, Spektrum Spezial 2/2009, p. 45 (on Newton).
%H Vincenzo Librandi, <a href="/A224445/b224445.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = numerator(r(n)) with the rationals (in lowest terms) r(n) = 2(1 - 2*sum(C(k-1)/4^(2*k),k=1..n), with the Catalan numbers C(n) = A000108(n).
%F r(n) gives the partial sums of the convergent series 2*sqrt(1 - 1/4), representing sqrt(3), with decimal expansion given in A002194.
%e The rationals r(n) are, for n=0..10:
%e 2, 7/4, 111/64, 887/512, 28379/16384, 227025/131072, 3632379/2097152, 29058999/16777216, 1859775507/1073741824, 14878203341/8589934592, 238051251025/137438953472.
%e The values for r(10^k), k = 0,..,3 are (Maple 10 digits):
%e 1.750000000, 1.732050812, 1.732050808, 1.732050808
%e This should be compared with sqrt(3) (Maple 10 digits): 1.732050808.
%t r[n_] := 2*(1 - 2*Sum[ CatalanNumber[k - 1]/4^(2*k), {k, 1, n}]); Table[r[n], {n, 0, 17}] // Numerator (* _Jean-François Alcover_, Apr 09 2013 *)
%Y Cf. A224446, A000108, A002194.
%K nonn,frac
%O 0,1
%A _Wolfdieter Lang_, Apr 09 2013
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