A270395 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r(k) = 1/Fibonacci(k+1).

2, 7, 57, 2713, 4918440, 22223269372702, 355194969748884199331083933, 896996605353313749663062291034129550464167047150212163, 710754225314643793883316602476833806192189702887005360976366457324682443530843343068467237316280025378530303

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

sqrt(1/3) = 1/2 + 1/(2*7) + 1/(3*57) + 1/(5*2713) + ...

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 = Sqrt[1/3]; 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(1/3)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 22 2016

Crossrefs

Cf. A269993, A000045, A020760.

Sequence in context: A294948 A178769 A121079 * A105183 A269994 A023364

Adjacent sequences: A270392 A270393 A270394 * A270396 A270397 A270398

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 22 2016

Status

approved