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A periodic recurrence relation is one containing a cycle of infinitely repeating elements.

Periodic Recurrence Relations

The Golden Recurrence Relation

The following recurrence relation has a cycle length of 5:

u2 = 1/(u0 u1f)

In this recurrence relation, φ is the Golden Ratio, (sqrt(5)+1)/2.  Some special properties of φ cause the cyclic repetition of this relation, in particular:

φ² - 1 = φ   and
φ² - φ = 1 

In terms of u0 and u1, I will calculate the next four elements of the sequence, to show that the u5 = u0.

u2 = u0-1 u1-f
u3 = u1-1 u2-f = u1-1 (u0-1 u1-f)-f = u0f u1f²-1 = u0f u1f
u4 = u2-1 u3-f = (u0-1 u1-f)-1 (u0f u1f)-f = u01-f² u1f-f² = u0-f u1-1
u5 = u3-1 u4-f = (u0f u1f)-1 (u0-f u1-1)-f = u0-f+f² u1-f+f = u0

5-cycle Relation

u2 = (u1+1)/u0  is a 5-cycle relation

Here's the proof:

c = (b+1)/a
d = (c+1)/b = ((b+1)/a+1)/b = (a+b+1)/(ab)
e = (d+1)/c = (((a+b+1)/(ab))+1)/((b+1)/a) = (a+1)/b
f = (e+1)/d = (((a+1)/b)+1)/((a+b+1)/(ab)) = a

6-cycle Relation

u2 = u1/u0 is a 5-cycle relation

Here's the proof:

c = b/a
d = c/b = (b/a)/b = 1/a
e = d/c = (1/a)/(b/a) = 1/b
f = e/d = (1/b)/(1/a) = a/b
g = f/e = (a/b)/(1/b) = a

Internet references

Mathworld -- Golden Ratio

Related pages in this website

Linear Recurrence Relation -- a closed form solution

Fibonacci Numbers -- whose successive ratios approach the Golden Ratio

Trig functions of Special Angles -- The golden ratio pops up here, too.

Formal Power Series

Generating Function

The webmaster and author of this Math Help site is Graeme McRae.