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A220397 A modified Engel expansion of sqrt(2). 5
1, 3, 6, 4, 2, 2, 4, 6, 23, 66, 108, 7738, 290, 9, 24, 32, 30, 4, 6, 3, 6, 24, 22, 2, 6, 20, 6, 9, 16, 5, 12, 4, 12, 22, 5, 8, 3, 6, 4, 2, 2, 4, 6, 2, 2, 2, 2, 13, 24, 2, 3, 4, 2, 2, 2, 2, 23, 44, 21, 40, 8, 14, 3, 6, 12, 10, 11, 30, 4, 4, 9, 4, 3, 4, 2, 16, 45, 46, 528 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

See A220393 for a 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..79.

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 = sqrt(2) - 1. Then a(1) = 1, a(2) = ceiling(1/x) and, for n >= 1, a(n+2) = 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..inf} 1/P(n) = 1 + 1/3 + 1/(3*6) + 1/(3*6*4) + 1/(3*6*4*2) + 1/(3*6*4*2*2) + .... For n >= 2, the error made in truncating this series to n terms is less than the n-th term.

CROSSREFS

Cf. A028254, A220335, A220336, A220337, A220338, A220393, A220394, A220395, A220396, A220398.

Sequence in context: A169842 A185588 A199737 * A021737 A011307 A243625

Adjacent sequences:  A220394 A220395 A220396 * A220398 A220399 A220400

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Dec 13 2012

STATUS

approved

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Last modified February 20 14:14 EST 2020. Contains 332078 sequences. (Running on oeis4.)