A270550 Denominators of r-Egyptian fraction expansion for -1 + golden ratio, where r(k) = 1/(2k-1).

2, 3, 29, 5427, 44351523, 2425185575687123, 28554057480843193512603851183810, 867562462760057302403082510579589337681874970064132683984477238

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..11

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

Example

golden ratio - 1 = 1/(1*2) + 1/(3*3) + 1/(5*29) + 1/(7*5427) + ...

Mathematica

r[k_] := 1/(2k-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}]

Prog

(PARI) r(k) = 1/(2*k-1);
f(k, x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x); );
a(k, x=(sqrt(5)-1)/2) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Apr 03 2016

Crossrefs

Cf. A269993, A005408, A094214.

Sequence in context: A141514 A078727 A350868 * A366477 A076977 A328833

Adjacent sequences: A270547 A270548 A270549 * A270551 A270552 A270553

Keyword

nonn,frac,easy

Author

Clark Kimberling, Apr 02 2016

Status

approved