A220396 A modified Engel expansion of the Euler-Mascheroni constant gamma.

2, 7, 18, 4, 2, 2, 3, 1466, 1464, 9, 24, 4, 2, 9, 104, 60, 8, 2, 3, 6, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 32, 30, 2, 13, 36, 6, 4, 3, 6, 6, 4, 4, 6, 2, 4, 6, 2, 4, 6, 9, 24, 4, 5, 8, 2, 2, 2, 2, 2, 3, 20

Offset

1,1

Comments

See A220393 for the definition of the modified Engel expansion of a positive real number. For further details see the Bala link.

Links

Table of n, a(n) for n=1..62.

Peter Bala, A modified Engel expansion

S. Crowley, Integral transforms of the harmonic sawtooth map, the Riemann zeta function, fractal strings, and a finite reflection formula, arXiv:1210.5652 [math.NT], 2012-2020.

Wikipedia, Engel Expansion

Formula

Let h(x) = x*(floor(1/x) + (floor(1/x))^2) - floor(1/x). Let x = gamma (see A001620). Then a(1) = 1 + floor(1/x) and, for n >= 1, a(n+1) = floor(1/h^(n-1)(x))*(1 + floor(1/h^(n)(x))).

Put P(n) = Product_{k = 1..n} a(k). Then we have the Egyptian fraction series expansion sqrt(2) = Sum_{n>=1} 1/P(n) = 1/2 + 1/(2*7) + 1/(2*7*18) + 1/(2*7*18*4) + 1/(2*7*18*4*2) + .... The error made in truncating this series to n terms is less than the n-th term.

Crossrefs

Cf. A001620, A053977, A220335, A220336, A220337, A220338, A220393, A220394, A220395, A220397, A220398.

Sequence in context: A013062 A013092 A174311 * A217253 A268837 A104310

Adjacent sequences: A220393 A220394 A220395 * A220397 A220398 A220399

Keyword

nonn,easy

Author

Peter Bala, Dec 13 2012

Status

approved