A270583 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r(k) = 1/(k+1).

1, 2, 4, 70, 6174, 45785878, 9941815425565254, 90769288470114014438337290256582, 35712953028973795679646888004332118441804994642382862330509857037

Offset

1,2

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(3) - 1 = 1/(2*1) + 1/(3*2) + 1/(4*4) + 1/(5*70) + ...

Mathematica

r[k_] := 1/(k+1); 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[3] - 1; Table[n[x, k], {k, 1, z}]

Crossrefs

Cf. A269993.

Sequence in context: A054920 A062061 A066974 * A018373 A292414 A053550

Adjacent sequences: A270580 A270581 A270582 * A270584 A270585 A270586

Keyword

nonn,frac,easy

Author

Clark Kimberling, Apr 03 2016

Status

approved