A270554 Denominators of r-Egyptian fraction expansion for e - 2, where r(k) = 1/(2k-1).

2, 2, 4, 89, 11084, 101449736, 10283734953162540, 146727957364669007427252221319539, 22046450568037230736928892396703267030745801074192006421746544107

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

e - 2 = 1/(1*2) + 1/(3*2) + 1/(5*4) + 1/(7*89) + ...

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 = E - 2; 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=exp(1)-2) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Apr 03 2016

Crossrefs

Cf. A269993, A005408, A001113.

Sequence in context: A050923 A326960 A067700 * A037010 A294184 A114695

Adjacent sequences: A270551 A270552 A270553 * A270555 A270556 A270557

Keyword

nonn,frac,easy

Author

Clark Kimberling, Apr 02 2016

Status

approved