|
%I #8 Dec 13 2012 10:40:22
%S 2,4,6,2,18,34,2,578,1154,2,665858,1331714,2,886731088898,
%T 1773462177794,2,1572584048032918633353218,3145168096065837266706434,
%U 2,4946041176255201878775086487573351061418968498178,9892082352510403757550172975146702122837936996354
%N A modified Engel expansion for 4*sqrt(2) - 5.
%C For a brief description of the modified Engel expansion of a real number see A220335.
%C Let p >= 2 be an integer and set Q(p) = (p - 1)*sqrt(p^2 - 1) - (p^2 - p - 1), so Q(3) = 4*sqrt(2) - 5. Iterating the identity Q(p) = 1/2 + 1/(2*(p+1)) + 1/(2*(p+1)*(2*p)) + 1/(2*(p+1)*(2*p))*Q(2*p^2-1) leads to a representation for Q(p) as an infinite series of unit fractions. The sequence of denominators of these unit fractions can be used to find the modified Engel expansion of Q(p). For further details see the Bala link. The present sequence is the case p = 3. For other cases see A220335 (p = 2), A220337 (p = 4) and A220338 (p = 5).
%H P. Bala, <a href="/A220335/a220335.pdf">A modified Engel expansion for certain quadratic irrationals</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]
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>
%F Define the map h(x) := floor(1/x)*(x*ceiling(1/x) - 1). Let x = 4*sqrt(2) - 5. Then a(1) = ceiling(1/x) and for n >= 2, a(n) = floor(1/h^(n-2)(x))*ceiling(1/h^(n-1)(x)), where h^(n)(x) denotes the n-th iterate of the map h(x), with the convention h^(0)(x) = x.
%F a(3*n+2) = 1/2*{2 + (1+sqrt(2))^(2^(n+1)) + (1-sqrt(2))^(2^(n+1))},
%F a(3*n+3) = {(1 + sqrt(2))^(2^(n+1)) + (1 - sqrt(2))^(2^(n+1))}, both
%F for n >= 0.
%F For n >= 0, a(3*n+1) = 2. For n >= 1, a(3*n+2) = 2*A001601(n)^2 and a(3*n+3) = 4*A001601(n)^2 - 2.
%F Recurrence equations:
%F For n >= 1, a(3*n+2) = 2*{a(3*n-1)^2 - 2*a(3*n-1) + 1} and
%F a(3*n+3) = 2*a(3*n+2) - 2.
%F Put P(n) = product(k = 0..n} a(k). Then we have the infinite Egyptian fraction representation 4*sqrt(2) - 5 = sum {n >=0} 1/P(n) = 1/2 + 1/(2*4) + 1/(2*4*6) + 1/(2*4*6*2) + 1/(2*4*6*2*18) + ....
%Y Cf. A001601, A028257, A220335 (p = 2), A220337 (p = 4), A220338 (p = 5).
%K nonn,easy
%O 1,1
%A _Peter Bala_, Dec 12 2012
| 