A270397 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r(k) = 1/Fibonacci(k+1).

2, 3, 6, 21, 411, 120274, 10572781147, 74407087111123560666, 5372512080606517833291366730287672914459, 41169436260792910821230360026041473906108740980452651576082359437785122898819171

Offset: 1

Author

Clark Kimberling, Mar 22 2016

Keywords

• nonn
• frac
• easy

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.

Examples

			sqrt(3) - 1 = 1/2 + 1/(2*3) + 1/(3*6) + 1/(5*21) + ...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..13
• Eric Weisstein's World of Mathematics, Egyptian Fraction.
• Index entries for sequences related to Egyptian fractions

Crossrefs

Cf. A269993, A000045, A160390.

Programs

Mathematica

r[k_] := 1/Fibonacci[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}]

PARI

r(k) = 1/fibonacci(k+1);
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=sqrt(3)-1) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 22 2016