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 A224445 Numerators of certain rationals approximating sqrt(3). 2
 2, 7, 111, 887, 28379, 227025, 3632379, 29058999, 1859775507, 14878203341, 238051251025, 1904410004001, 60941120098639, 487528960737109, 7800463371608019, 62403706972529847, 7987674492474125571, 63901395939775325733 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The corresponding denominators are given in A224446. The rationals r(n) are the partial sums of the series 2*sqrt(1 - 1/4) which represents sqrt(3). REFERENCES H. K. Strick, Geschichten aus der Mathematik, Spektrum Spezial 2/2009, p. 45 (on Newton). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 FORMULA 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). r(n) gives the partial sums of the convergent series 2*sqrt(1 - 1/4), representing sqrt(3), with decimal expansion given in A002194. EXAMPLE The rationals r(n) are, for n=0..10: 2, 7/4, 111/64, 887/512, 28379/16384, 227025/131072, 3632379/2097152, 29058999/16777216, 1859775507/1073741824, 14878203341/8589934592, 238051251025/137438953472. The values for r(10^k), k = 0,..,3 are (Maple 10 digits): 1.750000000, 1.732050812, 1.732050808, 1.732050808 This should be compared with sqrt(3) (Maple 10 digits): 1.732050808. MATHEMATICA 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 *) CROSSREFS Cf. A224446, A000108, A002194. Sequence in context: A072664 A352046 A045310 * A000157 A264999 A326940 Adjacent sequences: A224442 A224443 A224444 * A224446 A224447 A224448 KEYWORD nonn,frac AUTHOR Wolfdieter Lang, Apr 09 2013 STATUS approved

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Last modified December 9 12:57 EST 2023. Contains 367690 sequences. (Running on oeis4.)