A220394 A modified Engel expansion of exp(1).

1, 1, 2, 3, 4, 5, 8, 2, 10, 99, 20, 2, 2, 2, 2, 2, 2, 3, 6, 4, 8, 14, 2, 2, 4, 6, 10, 252, 81, 30, 28, 31, 60, 4, 6, 3, 4, 2, 2, 2, 2, 19, 54, 8, 6, 22, 63, 4, 2, 4, 6, 2, 2, 5, 12, 4, 2, 2, 2, 2, 6, 15, 10, 348, 172, 2, 2, 4, 6, 4, 30, 207, 220

Offset

1,3

Comments

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

The Engel expansion for exp(1) is the sequence of positive integers A000027.

Links

Table of n, a(n) for n=1..73.

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) where x = exp(1) - 2, then a(1) = a(2) = 1, a(3) = ceiling(1/x) and, for n >= 1, a(n+3) = 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 exp(1) = Sum_{n>=1} 1/P(n) = 1/1 + 1/1 + 1/2 + 1/(2*3) + 1/(2*3*4) + 1/(2*3*4*5) + 1/(2*3*4*5*8) + .... For n >= 3, the error made in truncating this series to n terms is less than the n-th term.

Crossrefs

Cf. A000027, A220335, A220336, A220337, A220338, A220393, A220395, A220396, A220397, A220398.

Sequence in context: A327324 A063948 A113929 * A082351 A122319 A123714

Adjacent sequences: A220391 A220392 A220393 * A220395 A220396 A220397

Keyword

nonn,easy

Author

Peter Bala, Dec 13 2012

Status

approved