A270375 Denominators of r-Egyptian fraction expansion for golden ratio - 1, where r = (1,1/4,1/9,1/16,...).

2, 3, 4, 10, 60, 4473, 23403582, 1295226544484567, 9611349042287513051537445592891, 89998772942534105602452834114784063917358549011796155052807149

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

tau - 1 = 1/2 + 1/(4*3) + 1/(9*4) + 1/(16*10) + ...

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 = GoldenRatio - 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(5)-1)/2) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

Crossrefs

Cf. A269993.

Sequence in context: A337536 A247204 A333891 * A049204 A045924 A244551

Adjacent sequences: A270372 A270373 A270374 * A270376 A270377 A270378

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 20 2016

Status

approved