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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)^xif 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 = 34/3 = 1.3333333...7/4 = 1.7511/7 = 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
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Here is another sequence you can make with the golden ratio.
....
...76, 123,....
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,
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