A270398 Denominators of r-Egyptian fraction expansion for golden ratio - 1, where r(k) = 1/Fibonacci(k+1).

2, 5, 19, 409, 110469, 11663878545, 142556979966838173805, 52663280147046053953610628211699561262739, 3562483323353729594120027219074361805521197466091774321103882341800358039125668071

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..12

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

Example

tau - 1 = 1/3 + 1/(2*5) + 1/(3*19) + 1/(5*409) + ...

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

Prog

(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(5)-1)/2) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 22 2016

Crossrefs

Cf. A269993, A000045, A094214.

Sequence in context: A055813 A119550 A119563 * A269997 A270547 A177875

Adjacent sequences: A270395 A270396 A270397 * A270399 A270400 A270401

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 22 2016

Status

approved