A270376 Denominators of r-Egyptian fraction expansion for 1/Pi, where r = (1, 1/4, 1/9, 1/16, ...).

4, 4, 20, 246, 150610, 28628772458, 4633718454684972107216, 32270052939985266099596531363945117655631355, 1531822012919710742180024988940181184501391371231597927081244972822399811859680584475111

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/Pi = 1/4 + 1/(4*4) + 1/(9*20) + 1/(16*246) + ...

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 = 1/Pi; 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=1/Pi) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

Crossrefs

Cf. A269993.

Sequence in context: A014433 A191366 A216164 * A323744 A205142 A072696

Adjacent sequences: A270373 A270374 A270375 * A270377 A270378 A270379

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 20 2016

Status

approved