A270405 - OEIS

A270405

Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/Fibonacci(k+1).

2, 2, 8, 99, 9153, 134325943, 17980902816814494, 336913028495678415812394391065577, 70730509948452535771375914216285436007372776802180962851035180747

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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

(1/2)^(1/3) = 1/2 + 1/(2*2) + 1/(3*8) + 1/(5*99) + ...

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 = (1/2)^(1/3); Table[n[x, k], {k, 1, z}]