A270401 Denominators of r-Egyptian fraction expansion for 1/e, where r(k) = 1/Fibonacci(k+1).

3, 15, 275, 306142, 119655359789, 11580087075793732204662, 149024368678486978900547818363959440890696944, 23508494642625759146052819702452314132546312046986774534693830017181700550956750715996250

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

1/e = 1/3 + 1/(2*15) + 1/(3*275) + ...

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

Crossrefs

Cf. A269993, A000045, A068985.

Sequence in context: A013354 A013356 A013353 * A270001 A138896 A090627

Adjacent sequences: A270398 A270399 A270400 * A270402 A270403 A270404

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 22 2016

Status

approved