The fractal sequence http://www.research.att.com/~njas/sequences/A054065 [Broken](adsbygoogle = window.adsbygoogle || []).push({});

is of interest because it provides permutations of the numbers 1-n such

that the decimal part of k*tau (k = {1,2,3,...n} is ordered from the

lowest possible value to the highest. For instance if n = 3 the

permutation is 2,1,3 since .2360 < .6180 <.8541. (3*(1 + sqrt5)/2 =

4.8541..). However, other than calculating the decimal parts for each

k*tau and sorting the list no algorithm is provided for obtaining the

proper permutation of the numbers 1-n.

I have the algorithm to share with you.

Let F(odd) be the highest fibonacci number less or equal to n and F(2b)

be the highest even fibonacci number less than n.

start with k = F(odd) since of the numbers 1-n, k=F(odd) provides the

lowest possible decimal part for k*tau. Now follow the two rules below

in the order listed to get the subsequent terms in the permutation:

1. add F(odd) if the resulting value is less than F(odd+2)

2. subtract F(2b)

If the result of the first operation yields a number higher then you want in the permutation omit it. It would have been a part of the permutation

if n was chosen to be as larger such as F(odd+2)-1, however, so you can choose

to leave it in if you prefer to increase the number of terms in the

permutation instead. But even if you omit it continue the algorithm as

though it was included to get the next n in the permutation.

I said the permutation was cyclic because in fact you could start

with any number up to n and follow the above 2 rules to get all n

numbers in the permutation as though you were going around a clock only

you started at 6 instead of at 1.

I hope I didn't bore you with this.

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# Construction of a cyclic sequence re the Golden Ratio

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