A073833 Numerators of b(n) where b(1) = 1, b(i) = b(i-1) + 1/b(i-1).

1, 2, 5, 29, 941, 969581, 1014556267661, 1099331737522548368039021, 1280590510388959061548230114212510564911731118541, 1726999038066943724857508638586386504281539279376091034086485112150121338989977841573308941492781

Offset: 1

Author

Alex Fink, Aug 12 2002

Keywords

• frac
• nonn

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

a(n) is also the numerator of b(n) where b(1) = 1 and b(2) = 2 and b(n) = b(n-2) + b(n-1) - (b(n-2)^2/b(n-1)) for n > 2. This has a geometric interpretation: One can prove, given two half lines starting at the center of a series of concentric circles, and a set of triangles each defined by the intersections of the two half lines with any given circle and one of the intersections of the rays with the next circle, that if the circles have radii specified by b(n), the triangle areas are all equal. - Sjoerd C. de Vries, Aug 13 2015

Examples

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

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

• Elijah M. Kin, Table of n, a(n) for n = 1..13
• Sjoerd C. de Vries, Mathematica file illustrating geometric application of the sequence
• Steven Finch, Popa's "Recurrent Sequences" and Reciprocity, arXiv:2412.11806 [math.CA], 2024. See p. 16.
• Clark Kimberling, Polynomials associated with reciprocation, JIS 12 (2009) 09.3.4.
• Eric Weisstein's World of Mathematics, Fractional Chromatic Number
• Eric Weisstein's World of Mathematics, Mycielski Graph

Crossrefs

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

Programs

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 *)
Numerator@RecurrenceTable[{b[n] == b[-2 + n] - b[-2 + n]^2/b[-1 + n] + b[-1 + n], b[1] == 1,
   b[2] == 2}, b, {n, 1, 10}] (* Sjoerd C. de Vries, Aug 13 2015 *)

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 */

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