A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quart., 11 (1973), 429--437.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....also works well for some rapidly growing sequences. Example 9.7. Doubly exponential sequences. Many recurrences are of the form a n 1 = a 2 n b n , 9. 56) where b n is much smaller than a 2 n (and may even depend on the a n for k # n, as in b n = a n or b n = a n 1 ) Aho and Sloane [3] found that surprisingly simple solutions to such recurrences can often be found. The basic idea is to reduce to approximate linearization by taking logarithms. We find that if a 0 is the given initial value, and a n 0 for all n, then the transformation u n = log a n , 9.57) # n = log(1 b n ....

....# k 0 . The new # for which (9.66) holds will then be defined in terms of x k 0 , x k 0 1 , In some situations the results presented above cannot be applied, but the basic method can still be extended. That is the case for the recurrence a n 1 = a n a n 1 1, a 0 , a 1 # 1 (9. 71) of [3]. The result is that a n is the nearest integer to # Fn # Fn 1 , 9.72) where # and # are positive constants, and the F k are the Fibonacci numbers. What matters is that the recurrence leads to doubly exponential (and regular) growth of a n . Example 15.3 shows how this principle can be ....

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quart., 11 (1973), 429--437.

....458330; 210066388901; Note that um is the number of binary search trees with height at most m Gamma 1 and (12) has been studied from this point of view. While no closed form solution to (12) is known, one can show that um = bK 2 m c = bK n 1 c where K is approximately 1.502837. See Aho and Sloane (1973) for a discussion of this and other nonlinear recurrences of the form x n 1 = x 2 n g n , where g n is a slowly growing function of n. 3. One approach to computing D(S; T ; Delta) begins by constructing tables of ancestry relations for S and T . It is easy to see how to construct such ....

Aho, A. V. and Sloane, N. J. A. (1973). Some doubly exponential sequences. Fibonacci Quarterly, 11 429--437.

.... z = r, r r, y h (z) is large only in a very small neighborhood of the real axis, where y h (z) g(z) a(z) d h (1 o(1) as h , 2.11) where g(z) and a(z) exp(b(z) are certain analytic functions. This result generalizes some earlier work on numerical recurrences by Aho and Sloane [3] and Reingold [33] The coefficients y h , n are studied by means of the Cauchy formula y h , n = 2p i 1 z = r y h (z) z n 1 dz . The radius r is chosen to make the value of the integral at z = r (where the maximum over the circle z = r is located) as small as possible, ....

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quart. 11 (1973), 429-437.

....we obtain enable us to determine the asymptotic behavior of the y h,n by expressing them as contour integrals and using the saddle point method. The key to the success of this method is the doubly exponential growth (1.11) of the y h (z) Equation (1. 11) generalizes the results of Aho and Sloane [2] about integer sequences satisfying nonlinear recurrences of the type x n 1 = x n 2 g n with g n 4 x n for n n 0 . Our results are related to the immense literature on the subject of rational iteration. See, for example, 3,4,8] Most of the papers in that area are concerned ....

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quart. 11 (1973), 429-437.

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A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quart., 11 (1973), 429--437.

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