A270584 Denominators of r-Egyptian fraction expansion for golden ratio - 1, where r(k) = 1/(k+1).

1, 3, 37, 1204, 21029921, 425355555167420, 439183524292095499600664584581, 240317442633783387248198509182959563857071128274317237128901, 1816763565571992723556609635427913847146292698536599340539742991592182627925499061514094793847919952134648005118828414904

Offset: 1

Author

Clark Kimberling, Apr 03 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

			tau - 1 = 1/(2*1) + 1/(3*3) + 1/(4*37) + 1/(5*1204) + ...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..11
• Eric Weisstein's World of Mathematics, Egyptian Fraction
• Index entries for sequences related to Egyptian fractions

Crossrefs

Cf. A269993.

Programs

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 = GoldenRatio - 1; Table[n[x, k], {k, 1, z}]