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%I #11 Jul 26 2021 01:48:14
%S 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,
%T 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,
%U 44,21,40,8,14,3,6,12,10,11,30,4,4,9,4,3,4,2,16,45,46,528
%N A modified Engel expansion of sqrt(2).
%C See A220393 for a definition of the modified Engel expansion of a positive real number. For further details see the Bala link.
%H Peter Bala, <a href="/A220393/a220393.pdf">A modified Engel expansion</a>
%H S. Crowley, <a href="http://arxiv.org/abs/1210.5652">Integral transforms of the harmonic sawtooth map, the Riemann zeta function, fractal strings, and a finite reflection formula</a>, arXiv:1210.5652 [math.NT], 2012-2020.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>
%F 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))).
%F 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 + 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.
%Y Cf. A028254, A220335, A220336, A220337, A220338, A220393, A220394, A220395, A220396, A220398.
%K nonn,easy
%O 1,2
%A _Peter Bala_, Dec 13 2012
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