I came across a pattern of numbers, while working on Fibonacci sequence. To my knowledge it appears to be the shortest one found yet.
To obtain this pattern we process the numbers in Fibonacci sequence in a special way. The process is,
1] Obtain square of each number from Fibonacci sequence. Make sure that the Fibonacci sequence has at least 13 numbers.(As the pattern that repeats is 12 units long)
2] Obtain the digital root of these squares and make a sequence of them such that each digital root has the same index as that of the corresponding Fibonacci number.
3] If you look at the pattern of these digital roots, you will find that it is like this, 1,1,4,9,7,1,7,9,4,1,1,9 which keeps repeating itself.
If we consider a Fibonacci sequence of fairly large length it is observed that this 12 number pattern repeats. Another known pattern related to Fibonacci series is 60 units in length; which considers right most digit of a Fibonacci number to generate the pattern.
Can we consider the pattern of these digital roots of squares of the Fibonacci numbers, as a new shortest pattern related to the Fibonacci numbers?