A086403 Numerators in continued fraction representation of (e-1)/(e+1).

1, 6, 61, 860, 15541, 342762, 8927353, 268163352, 9126481321, 347074453550, 14586253530421, 671314736852916, 33580323096176221, 1814008761930368850, 105246088515057569521, 6527071496695499679152, 430891964870418036393553, 30168964612425958047227862

Offset: 1

Author

Gary W. Adamson, Jul 18 2003

Keywords

• nonn

Examples

			a(4) = 860 = closest integer to[(e-1)/(e+1)]*A079165(4); = floor(860.0000292...) = 860. 860/1861 = [2, 6, 10, 14] = .462117141...; (e-1)/(e+1) = .462117157...
		

References

• Calvin C. Clawson, "Mathematical Mysteries", Perseus, 1999, p. 225.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..360

Crossrefs

Cf. A000108, A016825, A033184, A079165.

Programs

Maple

b:= proc(n) local i, q;
      q:= 0;
      for i to n do
        q:= 1/(q+4*(n-i)+2)
      od; q
    end:
a:= n-> numer(b(n)):
seq(a(n), n=1..20);  # Alois P. Heinz, Feb 03 2012
numtheory:-cfrac((exp(1)-1)/(exp(1)+1),50,'convergents'):
map(numer,convergents[2..-2]); # Robert Israel, Apr 26 2016

Mathematica

Numerator@ FromContinuedFraction@ ContinuedFraction[(E - 1)/(E + 1), #] & /@ Range[2, 19] (* Michael De Vlieger, Apr 26 2016 *)

Maxima

a(n):=(sum((2*n-2*k+1)!/((2*k-1)!*(n-2*k+1)!),k,1,(n+1)/2));
taylor(sinh((1-sqrt(1-4*x))/2)/sqrt(1-4*x),x,0,10); /* Vladimir Kruchinin, Apr 26 2016 */

Formula

Partial quotients in continued fraction representation of (e-1)/(e+1) are A016825: [2, 6, 10, 14, 18...], the convergents being: [2] = 1/2, [2, 6] = 6/13, [2, 6, 10] = 61/132...etc.; denominators are A079165 starting with n=1: 2, 13, 132, 1861, 33630, 741721, 19318376... 2. a(n) = closest integer to [(e-1)/(e+1)]*A079165(n), n>0

E.g.f.: sinh((1-sqrt(1-4*x))/2)/sqrt(1-4*x). - Vladimir Kruchinin, Apr 26 2016

a(n) = Sum_{k=1..(n+1)/2} (2*n-2*k+1)!/((2*k-1)!*(n-2*k+1)!). - Vladimir Kruchinin, Apr 26 2016

a(n) = -((-1)^n*sqrt(Pi/exp(1))*BesselI((2*n+1)/2, 1/2))/2 + (BesselK((2*n+1)/2, 1/2)*sinh(1/2))/sqrt(Pi), where BesselI(n,x) is the modified Bessel function of the first kind, BesselK(n,x) is the modified Bessel function of the second kind. - Ilya Gutkovskiy, Apr 26 2016

From Vaclav Kotesovec, Apr 27 2016: (Start)

a(n)/n! ~ BesselI(1/2, 1/2) * 2^(2*n-1) / sqrt(n).

a(n) ~ sinh(1/2) * 2^(2*n + 1/2) * n^n / exp(n).

(End)

Extensions

More terms from Alois P. Heinz, Feb 03 2012