A270521 Denominators of r-Egyptian fraction expansion for -1 + golden ratio, where r(k) = 1/k!.

2, 5, 10, 31, 359, 59268, 20093595288, 65918810162398275290, 1376774212248790880568204797925263397693, 867295417612705949534513217399066894914443846832286344139318038624107644813557

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

Table of n, a(n) for n=1..10.

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

Example

tau - 1 = 1/(1*2) + 1/(2*5) + 1/(6*10) + 1/(24*31) + ...

Mathematica

r[k_] := 1/k!; 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 + GoldenRatio; Table[n[x, k], {k, 1, z}]

Prog

(PARI) r(k) = 1/k!;
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 31 2016

Crossrefs

Cf. A269993, A000142, A094214.

Sequence in context: A155580 A226963 A018386 * A089073 A343167 A376849

Adjacent sequences: A270518 A270519 A270520 * A270522 A270523 A270524

Keyword

nonn,frac,easy,changed

Author

Clark Kimberling, Mar 30 2016

Status

approved