The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A270372 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r = (1, 1/4, 1/9, 1/16, ...). 1

%I

%S 2,4,8,66,2776,20101656,1227318932297655,

%T 8216049453479522437439630860819,

%U 474082010892842884364582298006064172482079224559365990598026496

%N Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r = (1, 1/4, 1/9, 1/16, ...).

%C Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ..., the r-Egyptian fraction for x.

%C See A269993 for a guide to related sequences.

%H Clark Kimberling, <a href="/A270372/b270372.txt">Table of n, a(n) for n = 1..12</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>

%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>

%e sqrt(1/3) = 1/2 + 1/(4*4) + 1/(9*2) + 1/(16*3) + 1/(25*7) + ...

%t r[k_] := 1/k^2; f[x_, 0] = x; z = 10;

%t n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]

%t f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]

%t x = Sqrt[1/3]; Table[n[x, k], {k, 1, z}]

%o r(k) = 1/k^2;

%o f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););

%o a(k, x=sqrt(1/3)) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 21 2016

%Y Cf. A269993.

%K nonn,frac,easy

%O 1,1

%A _Clark Kimberling_, Mar 20 2016

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 13 22:34 EDT 2020. Contains 336465 sequences. (Running on oeis4.)