More fun with the golden ratio (Score 1)

Here is another sequence you can make with the golden ratio.

phi = 1.6180339887....

phi^1 - (1/phi)^1 = 1

phi^2 + (1/phi)^2 = 3

phi^3 - (1/phi)^3 = 4

phi^4 + (1/phi)^4 = 7

phi^5 - (1/phi)^5 = 11 ....

if x = even then add phi^x + (1/phi)^x
if x = odd then subtract phi^x - (1/phi)^x

it will generate the sequence 1,3,4,7,11, ...76, 123,....

now if you divide the a number in the sequence by the previous number you approach the limit phi

3/1 = 3
4/3 = 1.3333333...
7/4 = 1.75
11/7 = 1.5714...
18/11 = 1.6363...
29/11 = 1.6111111...
.
.
.
123/76 = 1.61842...

I wonder how many other sequence may be related to phi