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Of the sequences on page 594 , (a) through (d) as well as (f) fail every single one of the tests, (e) fails only the serial test, while (g) and (h) pass all the tests.
And it is perhaps notable that while such activities as music and thought are often discussed, exploration essentially never is.
But for example x a x + b and x 1/(a + b x) both give results just involving powers, while x Sqrt[a x + b] sometimes yields trigonometric functions, as on page 915 .
But while Leibniz considered the possibility of checking his descriptions by machine, he apparently did not imagine setting up the analog of a computation in which something is explicitly generated from input that has been given.
Some concentrate on specific nearby stars, while others look at the whole sky, or test the stream of data from all observations at a particular radio telescope, sometimes scanning for repetitive trains of pulses rather than single frequencies.
The color of the element at position n is given by 2 - (Floor[(n + 1) GoldenRatio] - Floor[n GoldenRatio]) (see page 904 ), while the position of the k th white element is given by the so-called Beatty sequence Floor[k GoldenRatio] .
On a single plant different leaves can have somewhat different shapes—sometimes for example those lower on a tree are smoother, while those higher are pointier.
The rest of the boundary consists of a sequence of algebraic curves, with almost imperceptible changes in slope in between; the first corresponds to {0, 0, 0, 1, 0, 1, 0, 1, …} , while subsequent ones correspond to {0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, …} , {0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, …} , etc.
But while the system was successful with a few simple theorems the searches it had to do rapidly became far too slow.
In the limit of an infinite number of steps the filling fraction tends to 1, while the region left unfilled has a fractal dimension of about 1.3057.