A233770 -id:A233770 - cp's OEIS frontend

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

Alex Fink, Aug 12 2002

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

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

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.

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

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

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 *)

{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 */

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

A232975 Decimal expansion of lim_{n -> infinity} (1/n)exp(2(b(n)^2-2n)) where b(i) = b(i-1) + 1/b(i-1) for i >= 2, b(1) = 1 (see A073833).

Original entry on oeis.org

5, 7, 4, 8, 1, 0, 2, 7, 4, 6, 7, 1, 7, 8, 5, 0, 6, 5, 2, 9, 6, 3, 4, 7, 8, 4, 6, 3, 5, 3, 0, 9, 2, 8, 0, 0, 7, 6, 0, 9, 8, 7, 8, 4, 5, 5, 2, 0, 8, 9, 0, 6, 1, 9, 0, 1, 5, 1, 5, 5, 4, 7, 8, 6, 6, 6, 4, 5, 5, 7, 3, 4, 1, 9, 9, 1, 6, 2, 5, 9, 1, 2, 6, 9, 5, 8, 7

Offset: 0

Jon E. Schoenfield and N. J. A. Sloane, Dec 07 2013

A332396 Decimal expansion of lim_{n->infinity} (b(n)/n - (n - log(n))/4) where {b(n)} is the real-valued sequence defined by b(1)=1 and b(n+1) = b(n) + sqrt(b(n)) for n > 0.

Original entry on oeis.org

6, 7, 5, 1, 7, 7, 4, 4, 2, 4, 5, 8, 5, 5, 7, 1, 3, 9, 8, 1, 3, 2, 8, 5, 6, 2, 5, 0, 7, 5, 8, 2, 7, 6, 3, 3, 6, 8, 4, 0, 7, 3, 1, 5, 8, 9, 8, 9, 0, 5, 1, 5, 9, 1, 4, 0, 1, 1, 3, 0, 8, 1, 0, 8, 5, 9, 1, 5, 1, 9, 7, 2, 9, 4, 9, 8, 4, 0, 2, 3, 7, 9, 3, 5, 3, 3, 7

Offset: 0

Jon E. Schoenfield, Feb 10 2020

Consider the "add the square root" sequence {b(n)} starting with 1, i.e., b(1)=1, b(n+1) = b(n) + sqrt(b(n)) for n > 0: at each step, the next term is obtained simply by adding the current term to its square root, so the sequence begins {1, 2, 2 + sqrt(2), 2 + sqrt(2) + sqrt(2 + sqrt(2)), ...}, i.e., {1, 2, 3.414213..., 5.261972..., ...}.
The 1st, 10th, 100th, and 1000th terms are
    b(10^0) = 1.0
    b(10^1) = 25.956...
    b(10^2) = 2452.850...
    b(10^3) = 248949.869...
    b(10^4) = 24983729.385...
    b(10^5) = 2499779700.562...
Since the difference between successive terms is b(n+1) - b(n) = sqrt(b(n)) and the similar differential equation (d/dx)f(x) = sqrt(f(x)) is satisified by the function f(x) = (1/4)*x^2, it is perhaps not surprising that lim_{n->infinity} b(n)/n^2 = 1/4. More precisely, it can be shown that, as n increases, b(n) approaches
   (1/4)*n^2
   + u*n
   + u^2 - u/2
   + (-(1/2)*u^2 + u/4 - 1/96)/n
   + ((1/3)*u^3 + 0*u^2 + (-5/48)*u + 7/576)/n^2
   + ...
where u = -(1/4)*log(n) + c
  and c = 0.675177442458557139813285625075...
It follows that lim_{n->infinity} (b(n)/n - (n - log(n))/4) = c.
If we were to define {b(n)} instead as b(1)=1, b(n+1) = b(n) + 1/b(n) for n > 0 (i.e., "add the reciprocal" rather than "add the square root"), we would obtain the rational-valued sequence {b(n)} = {1, 2, 5/2, 29/10, 941/290, ...} (see A073833 and, for a constant arising from that sequence, A233770).

			0.67517744245855713981328562507582763368407315898905...

Steven Finch, Popa's "Recurrent Sequences" and Reciprocity, arXiv:2412.11806 [math.CA], 2024. See p. 16.

Cf. A073833, A233770.

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

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A232975 Decimal expansion of lim_{n -> infinity} (1/n)exp(2(b(n)^2-2n)) where b(i) = b(i-1) + 1/b(i-1) for i >= 2, b(1) = 1 (see A073833).

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A332396 Decimal expansion of lim_{n->infinity} (b(n)/n - (n - log(n))/4) where {b(n)} is the real-valued sequence defined by b(1)=1 and b(n+1) = b(n) + sqrt(b(n)) for n > 0.

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cp's OEIS Frontend

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

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A232975 Decimal expansion of lim_{n -> infinity} (1/n)*exp(2*(b(n)^2-2n)) where b(i) = b(i-1) + 1/b(i-1) for i >= 2, b(1) = 1 (see A073833).

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A332396 Decimal expansion of lim_{n->infinity} (b(n)/n - (n - log(n))/4) where {b(n)} is the real-valued sequence defined by b(1)=1 and b(n+1) = b(n) + sqrt(b(n)) for n > 0.

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A232975 Decimal expansion of lim_{n -> infinity} (1/n)exp(2(b(n)^2-2n)) where b(i) = b(i-1) + 1/b(i-1) for i >= 2, b(1) = 1 (see A073833).