

A220398


A modified Engel expansion of the golden ration 1/2*(1 + sqrt(5)).


6



1, 2, 5, 8, 3, 4, 4, 6, 2, 162, 322, 2, 51842, 103682, 2, 5374978562, 10749957122, 2, 57780789062419261442, 115561578124838522882, 2, 6677239169351578707225356193679818792962, 13354478338703157414450712387359637585922, 2
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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..24.
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 = 1/2*(1 + sqrt(5))  1. Then a(1) = 1, a(2) = ceiling(1/x) and, for n >= 1, a(n+2) = floor(1/h^(n1)(x))*{1 + floor(1/h^(n)(x))}.
Recurrence equations: For n >= 3, a(3*n) = 2. For n >= 4 we have a(3*n+2) = 2*a(3*n+1)  2 and a(3*n+1) =2*(a(3*n2)  1)^2.
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 + 1/2 + 1/(2*5) + 1/(2*5*8) + 1/(2*5*8*3) + 1/(2*5*8*3*4) + .... For n >= 2, the error made in truncating this series to n terms is less than the nth term.


CROSSREFS

Cf. A028259, A220335, A220336, A220337, A220338, A220393, A220394, A220395, A220396, A220397.
Sequence in context: A220337 A198545 A296430 * A200225 A258749 A056886
Adjacent sequences: A220395 A220396 A220397 * A220399 A220400 A220401


KEYWORD

nonn,easy


AUTHOR

Peter Bala, Dec 13 2012


STATUS

approved



