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A073833 Numerators of b(n) where b(1) = 1, b(i) = b(i-1) + 1/b(i-1). 5
1, 2, 5, 29, 941, 969581, 1014556267661, 1099331737522548368039021, 1280590510388959061548230114212510564911731118541, 1726999038066943724857508638586386504281539279376091034086485112150121338989977841573308941492781 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is also the numerator of the fractional chromatic number of the Mycielski graph M_n. - Eric W. Weisstein, Mar 05 2011

It appears that lim n -> infinity (1/n)*exp(2*(b(n)^2-2n)) = c1 = 0.57...... - Benoit Cloitre, Oct 16 2002

c1 = 0.574810274671785...; see A232975. - Jon E. Schoenfield, Nov 30 2013

b(n)^2 = t/2 + u + (u-1/2)/t + (-u^2+2*u-11/12)/t^2 + (4*u^3/3-5*u^2+17*u/3-65/36)/t^3 + ... where t=4*n, u=(log n)/2+c, and c=-0.2768576248625765389364372...; see A233770. - Jon E. Schoenfield, Dec 15 2013

a(n) is also the numerator of b(n) where b(0) = b(1) = 1 and b(n) = (b(n-1)^2 + b(n-2)^2) / b(n-2) for n>1 where the denominator of b(n) is partial products of A073834. - Michael Somos, Aug 16 2014

REFERENCES

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 187.

D. J. Newman, A Problem Seminar, Springer; see Problem #60.

J. H. Silverman, The arithmetic of dynamical systems, Springer, 2007, see p. 113 Table 3.1

LINKS

Table of n, a(n) for n=1..10.

Eric Weisstein's World of Mathematics, Fractional Chromatic Number

FORMULA

a(n) = a(n-1)^2 + A073834(n-1)^2; A073834(n) = a(n-1) * A073834(n-1). - Franklin T. Adams-Watters, Aug 04 2008

0 = a(n)^2*(a(n+1) - a(n)^2) - (a(n+2) - a(n+1)^2) for all n>0. - Michael Somos, Aug 16 2014

EXAMPLE

1, 2, 5/2, 29/10, 941/290, 969581/272890, 1014556267661/264588959090, 1099331737522548368039021/268440386798659418988490, ...

MATHEMATICA

f[n_]:=n+1/n; Prepend[Numerator[NestList[f, 2, 9]], 1] (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)

Numerator[NestList[# + 1/# &, 1, 10]] (* Eric W. Weisstein, Mar 05 2001 *)

a[ n_] := If[ n<1, 0, If[ n<3, n, With[{x = a[n-2]^2, y = a[n-1]}, y y + x y - x x]]]; (* Michael Somos, Aug 16 2014 *)

PROG

(PARI) {a(n) = if( n<1, 0, if( n<3, n, my(x = a(n-2)^2, y = a(n-1)); y^2 + x*y -x^2))}; /* Michael Somos, Mar 05 2012 */

CROSSREFS

See A073834 for denominators. See A232975 for c1; see A233770 for c.

Sequence in context: A059784 A000283 A121910 * A229918 A179554 A086383

Adjacent sequences:  A073830 A073831 A073832 * A073834 A073835 A073836

KEYWORD

frac,nonn

AUTHOR

Alex Fink (finks(AT)telus.net), Aug 12 2002

STATUS

approved

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Last modified December 21 04:56 EST 2014. Contains 252293 sequences.