MesMath

The fibonacci numbers
         

Anjusha M Achuthakumar

The fibonacci numbers occur in the sums of ''shallow'' diagonals in Pascal's triangle. The fibonacci numbers can be found in different ways in the sequence of binary strings.

·  The number of binary strings of length n without consequetive 1s is a fibonacci number F_n +2 . For example, out of the 16 binary strings of length 4, there are F ₆ = 8 without consecutive 1s they are 0000, 1000, 0100, 0010, 1010, 0001, 1001 and 0101.By symmetry, the number of strings of length n without consecutive 0s is also F_n +2.

·  The number of binary strings of length n without an odd number of consecutive 1s is the Fibonacci number F_n+1 . For example, out of the 16 binary strings of length 4, there are F ₅ = 5 without an odd number of consecutive 1s – they are 0000, 0011, 0110, 1100, 1111.

·  The number of binary strings of length n without an even number of consecutive 0s or 1s is 2F_n . For example, out of the 16 binary strings of length 4, there are 2F ₄ = 6 without an even number of consecutive 0s or 1s – they are 0001, 1000, 1110, 0111, 0101, 1010