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A220394 A modified Engel expansion of exp(1). 5
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 (list; graph; refs; listen; history; text; internal format)
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 natural numbers A000027.

LINKS

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

P. 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]

Wikipedia, Engel Expansion

FORMULA

Let h(x) =  x*{floor(1/x) + (floor(1/x))^2} - floor(1/x).

Let 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..inf} 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

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Last modified February 24 10:37 EST 2020. Contains 332209 sequences. (Running on oeis4.)