A270372 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r = (1, 1/4, 1/9, 1/16, ...).

2, 4, 8, 66, 2776, 20101656, 1227318932297655, 8216049453479522437439630860819, 474082010892842884364582298006064172482079224559365990598026496

Offset

1,1

Comments

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.

See A269993 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 1..12

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

Example

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

Mathematica

r[k_] := 1/k^2; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt[1/3]; Table[n[x, k], {k, 1, z}]

Prog

(PARI) r(k) = 1/k^2;
f(k, x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x); );
a(k, x=sqrt(1/3)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

Crossrefs

Cf. A269993.

Sequence in context: A167182 A058345 A093843 * A018507 A018523 A261714

Adjacent sequences: A270369 A270370 A270371 * A270373 A270374 A270375

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 20 2016

Status

approved