1,1

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.

Clark Kimberling, Table of n, a(n) for n = 1..11

Eric Weisstein's World of Mathematics, Egyptian Fraction

1/Pi = 1/4 + 1/(2*8) + 1/(3*58) + 1/(5*3187) + ...

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/Pi; Table[n[x, k], {k, 1, z}]

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

Cf. A269993, A000045, A049541.

Sequence in context: A063083 A349859 A206346 * A269998 A335527 A303284

Adjacent sequences: A270396 A270397 A270398 * A270400 A270401 A270402

nonn,frac,easy

Clark Kimberling, Mar 22 2016

approved