Interesting pattern in Fibonacci Series

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Hi everyone, in this blog post I will be describing an interesting pattern formed by fibonacci numbers. Fibonacci number or fibonacci sequence is a sequence of numbers such that a given number is the sum of the previous 2 numbers.

In mathematical terms, F(n) = F(n-1) + F(n-2).

Fibonacci sequence is as follows - 0,1,1,2,3,5,8,13,21,34,55,89,144,233, 377, 610, 987....etc. There are plenty of unknown sequence and patterns in fibonacci sequence. If we carefully look the series, we ourself can figure out many unknown patterns. Here, I am specifying one of those unknown patterns in fibonacci sequence.

 Suppose we took the number 610.

It can be represented in the following ways:-

  1. 610 = 610 * 1 + 377 * 0 
  2. 610 = 377 * 1 + 233 * 1
  3. 610 =  233 * 2 + 144 * 1
  4. 610 = 144 * 3 + 89 * 2
  5. 610 = 89 * 5 + 55 * 3
  6. 610 = 55 * 8 + 34 * 5
  7. 610 = 34 * 13 + 21 * 8
  8. 610 = 21 * 21 + 13 * 13
  9. 610 = 13 * 34 + 8 * 21
  10. 610 = 8 * 55 + 5 * 34
  11. 610 = 5 * 89 + 3 * 55
  12. 610 = 3 * 144 + 2 * 89
  13. 610 = 2 * 233 + 1 * 144
  14. 610 = 1 * 377 + 1 * 233
  15. 610 = 1 * 610 + 0 * 377
In this list, if we see the factors of product terms, that factors are also in fibonacci sequence. Here, we can see that 610 can be represented in 8 unique ways using fibonacci  numbers. Those products from 9 -15 line number are repetitions. 610 is the 16th number in fibonacci sequence. So, In general, a number F(k) can be represented in k/2 unique ways (using fibonacci numbers). It doesn't matter if F(k) is even or odd. In general, a fibonacci number F(k) can be written as F(k) = (F(k+1-i) * F(i+1) ) + (F(k-i) * F(i)) for all i>0 and i<k.

Also one of the pattern you can see here is in 8th line we can see 610 is the sum of 2 square terms. This is true in the case of every even fibonacci number.

So, we can deduce one more formula, F(k) = ((F(k/2)) ^2)  + ((F((k/2)+1)) ^ 2).

So, as I said earlier, if we analyze the patterns in fibonacci, there would be plenty of it. 

I hope this post gave you a clear idea about one of the unrecognized fibonacci patterns. Please comment below if you have any doubts or suggestions. All comments will be responded within one day. 

Thank you for reading the post.









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