A270549 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r(k) = 1/(2k-1).

2, 2, 4, 10, 102, 9988, 462079550, 1246459580018549814, 3451767175159069042246539740614797183, 16047263805335625632784779620610026996218698731392917143951229224582015756

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..13

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

Example

sqrt(3) - 1 = 1/(1*2) + 1/(3*2) + 1/(5*4) + 1/(7*10) + ...

Mathematica

r[k_] := 1/(2k-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(3) - 1; Table[n[x, k], {k, 1, z}]

Prog

(PARI) r(k) = 1/(2*k-1);
f(k, x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x); );
a(k, x=sqrt(3)-1) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Apr 03 2016

Crossrefs

Cf. A269993, A005408, A160390.

Sequence in context: A322924 A173100 A092635 * A321204 A196856 A196707

Adjacent sequences: A270546 A270547 A270548 * A270550 A270551 A270552

Keyword

nonn,frac,easy

Author

Clark Kimberling, Apr 02 2016

Status

approved