#FIBONACCI SEQUENCE GOLDEN RATIO SERIES#

There is a fence with n posts, each post can be painted with one of the k colors. If there happens to be a way to compute the nth sequence in constant time, that would be fantastic. One way that this could be achieved is by using the (n-1)th sequence to compute the nth sequence. Ideally, to compute the nth sequence would require time proportional to n. I.e., I need a way to efficiently compute the following sequences: In order to solve the problem, I need a way to compute the diagonals shown above in a computationally efficient way. I realized that the underlying structure IS the Fibonacci sequence. When the first number to the right of the 1 in any row is a prime number, all numbers in that row are divisible by that prime number It can be used to find combinations in probability problems (if, for instance, you pick any two of five items, the number of possible combinations is 10, found by looking in the second place of the fifth row. The horizontal rows represent powers of 11 (1, 11, 121, 1331, 14641) for the first 5 rows, in which the numbers have only a single digit.Īdding any two successive numbers in the diagonal 1-3-6-10-15-21-28… results in a perfect square (1, 4, 9, 16, etc.) Horizontal rows add to powers of 2 (i.e., 1, 2, 4, 8, 16, etc.) Pascal’s triangle has many unusual properties and a variety of uses: The numbers on diagonals of the triangle add to the Fibonacci series, as shown below. Every number below in the triangle is the sum of the two numbers diagonally above it to the left and the right, with positions outside the triangle counting as zero. Pascal’s Triangle, developed by the French Mathematician Blaise Pascal, is formed by starting with an apex of 1.

#FIBONACCI SEQUENCE GOLDEN RATIO SERIES#

The Fibonacci Series is found in Pascal’s Triangle.

Fibonacci sequence golden ratio