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:

u_{2} = 1/(u_{0} u_{1}^{f})

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 u_{0} and u_{1}, I will calculate the next four
elements of the sequence, to show that the u_{5} = u_{0}.

u_{2} = u_{0}^{-1} u_{1}^{-f}

u_{3} = u_{1}^{-1} u_{2}^{-f} = u_{1}^{-1}
(u_{0}^{-1} u_{1}^{-f})^{-f} = u_{0}^{f} u_{1}^{f²-1} =
u_{0}^{f} u_{1}^{f}

u_{4} = u_{2}^{-1} u_{3}^{-f} = (u_{0}^{-1} u_{1}^{-f})^{-1}
(u_{0}^{f} u_{1}^{f})^{-f} = u_{0}^{1-f²} u_{1}^{f-f²} =
u_{0}^{-f} u_{1}^{-1}

u_{5} = u_{3}^{-1} u_{4}^{-f} = (u_{0}^{f} u_{1}^{f})^{-1}
(u_{0}^{-f} u_{1}^{-1})^{-f} = u_{0}^{-f+f²} u_{1}^{-f+f} =
u_{0}

### 5-cycle Relation

u_{2} = (u_{1}+1)/u_{0} 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

u_{2} = u_{1}/u_{0} 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.