A224446 Denominators of certain rationals approximating sqrt(3).

1, 4, 64, 512, 16384, 131072, 2097152, 16777216, 1073741824, 8589934592, 137438953472, 1099511627776, 35184372088832, 281474976710656, 4503599627370496, 36028797018963968, 4611686018427387904, 36893488147419103232

Offset

0,2

Comments

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

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

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) = 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).

Example

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

Maple

A224446 := proc(n) (x/(exp(x)-1))^(3/2)*exp(x/2);
-pochhammer(1/2, n-1)*coeff(series(%, x, n+2), x, n); denom(%) end:
seq(A224446(n), n=0..17); # Peter Luschny, Apr 05 2014

Mathematica

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 *)

Crossrefs

Cf. A224445, A000108, A002194.

Sequence in context: A053959 A343725 A195800 * A128782 A353453 A318779

Adjacent sequences: A224443 A224444 A224445 * A224447 A224448 A224449

Keyword

nonn,frac

Author

Wolfdieter Lang, Apr 09 2013

Status

approved