A145051 - OEIS

A145051

Numerator of the first convergent to sqrt(n) using the recursion x = (n/x + x)/2.

1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21, 11, 23, 12, 25, 13, 27, 14, 29, 15, 31, 16, 33, 17, 35, 18, 37, 19, 39, 20, 41, 21, 43, 22, 45, 23, 47, 24, 49, 25, 51, 26, 53, 27, 55, 28, 57, 29, 59, 30, 61, 31, 63, 32, 65, 33, 67, 34, 69, 35, 71, 36, 73, 37, 75

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OFFSET

1,2

COMMENTS

This is the same as A026741 without the first 2 terms in A026741. The link describes the experimental derivation of the generating function.

From Jaroslav Krizek, May 28 2010: (Start)

Numerators of arithmetic means of the first n positive integers for n >= 1.

See A040001 - denominators of arithmetic means of the first n positive integers.

a(n) = A026741(n+1) = A000217(n) * A040001(n) / n. (End)

Minimum number of line segments to draw into a circle to partition the circle into n+1 congruent circular sectors, i.e., minimum number of straight cuts required to cut a circular cake into n+1 equal slices. - Felix Fröhlich, Sep 01 2015

LINKS

Table of n, a(n) for n=1..74.

Cino Hilliard, Roots by Recursion [Broken link]

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

FORMULA

From Paul Barry, Nov 22 2009: (Start)

G.f.: x*(1 + 3*x - x^3)/(1 - x^2)^2.

a(n+1) = (n + 2)*(3 - (-1)^n)/4;

a(n+1) = Sum_{k=0..n, if(k=floor(n/2) or k=floor((n+1)/2),1,0)*(k+1)}. (End)

E.g.f.: ((x + 2)*cosh(x) + (2*x + 1)*sinh(x) - 2)/2. - Stefano Spezia, Apr 04 2024

EXAMPLE

n=1, x=1; x = (1/1+1)/2 = 1/1;