A270373 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r = (1, 1/4, 1/9, 1/16, ...).

3, 4, 7, 25, 5546, 36482088, 14423934280776257, 1969937215073991451613042447271867, 3160555685801520768089757205744771458914199650397475324265981061618

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

sqrt(2) - 1 = 1/3 + 1/(4*4) + 1/(9*7) + 1/(16*25) + ...

Mathematica

r[k_] := 1/k^2; 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[2] - 1; Table[n[x, k], {k, 1, z}]

Prog

(PARI) r(k) = 1/k^2;
f(k, x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x); );
a(k, x=sqrt(2)-1) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

Crossrefs

Cf. A269993.

Sequence in context: A145593 A042037 A041091 * A117764 A113874 A007677

Adjacent sequences: A270370 A270371 A270372 * A270374 A270375 A270376

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 20 2016

Status

approved