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

%I A073833 #81 Feb 16 2025 08:32:46
%S A073833 1,2,5,29,941,969581,1014556267661,1099331737522548368039021,
%T A073833 1280590510388959061548230114212510564911731118541,
%U A073833 1726999038066943724857508638586386504281539279376091034086485112150121338989977841573308941492781
%N A073833 Numerators of b(n) where b(1) = 1, b(i) = b(i-1) + 1/b(i-1).
%C A073833 a(n) is also the numerator of the fractional chromatic number of the Mycielski graph M_n. - _Eric W. Weisstein_, Mar 05 2011
%C A073833 It appears that lim_{n->infinity} (1/n)*exp(2*(b(n)^2-2n)) = c1 = 0.57...... - _Benoit Cloitre_, Oct 16 2002
%C A073833 c1 = 0.574810274671785...; see A232975. - _Jon E. Schoenfield_, Nov 30 2013
%C A073833 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
%C A073833 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
%C A073833 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
%D A073833 H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 187.
%D A073833 D. J. Newman, A Problem Seminar, Springer; see Problem #60.
%D A073833 J. H. Silverman, The arithmetic of dynamical systems, Springer, 2007, see p. 113 Table 3.1.
%H A073833 Elijah M. Kin, <a href="/A073833/b073833.txt">Table of n, a(n) for n = 1..13</a>
%H A073833 Sjoerd C. de Vries, <a href="/A073833/a073833.nb">Mathematica file illustrating geometric application of the sequence</a>
%H A073833 Steven Finch, <a href="https://arxiv.org/abs/2412.11806">Popa's "Recurrent Sequences" and Reciprocity</a>, arXiv:2412.11806 [math.CA], 2024. See p. 16.
%H A073833 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Kimberling/kimberling56.html">Polynomials associated with reciprocation</a>, JIS 12 (2009) 09.3.4.
%H A073833 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FractionalChromaticNumber.html">Fractional Chromatic Number</a>
%H A073833 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MycielskiGraph.html">Mycielski Graph</a>
%F A073833 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
%F A073833 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
%e A073833 1, 2, 5/2, 29/10, 941/290, 969581/272890, 1014556267661/264588959090, 1099331737522548368039021/268440386798659418988490, ...
%t A073833 f[n_]:=n+1/n;Prepend[Numerator[NestList[f,2,9]],1] (* _Vladimir Joseph Stephan Orlovsky_, Nov 19 2010 *)
%t A073833 Numerator[NestList[# + 1/# &, 1, 10]] (* _Eric W. Weisstein_, Mar 05 2001 *)
%t A073833 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 *)
%t A073833 Numerator@RecurrenceTable[{b[n] == b[-2 + n] - b[-2 + n]^2/b[-1 + n] + b[-1 + n], b[1] == 1,
%t A073833    b[2] == 2}, b, {n, 1, 10}] (* _Sjoerd C. de Vries_, Aug 13 2015 *)
%o A073833 (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 */
%Y A073833 See A073834 for denominators. See A232975 for c1; see A233770 for c.
%K A073833 frac,nonn
%O A073833 1,2
%A A073833 _Alex Fink_, Aug 12 2002