A224445 Numerators of certain rationals approximating sqrt(3).

2, 7, 111, 887, 28379, 227025, 3632379, 29058999, 1859775507, 14878203341, 238051251025, 1904410004001, 60941120098639, 487528960737109, 7800463371608019, 62403706972529847, 7987674492474125571, 63901395939775325733

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_{k=1..n} C(k-1)/4^(2*k)), 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