A270519 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/k!.

3, 7, 18, 217, 21586, 132830816, 8232750479147118, 8738244742575919521189548340591, 28575128242342620144630216663972970082807062570299713849045286

Offset: 1

Author

Clark Kimberling, Mar 30 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(2) - 1 = 1/(1*3) + 1/(2*7) + 1/(6*18) + 1/(24*217) + ...
		

Links

• Eric Weisstein's World of Mathematics, Egyptian Fraction
• Index entries for sequences related to Egyptian fractions

Crossrefs

Cf. A269993, A000142, A188582.

Programs

Mathematica

r[k_] := 1/k!; 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[2] - 1; Table[n[x, k], {k, 1, z}]

PARI

r(k) = 1/k!;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=sqrt(2)-1) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 31 2016