Fibonacci triangle or the Hosoya's triangle is a triangular arrangement of numbers (like Pascal's triangle

Pascal's triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...

) based on the Fibonacci number

Fibonacci number

In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....

s. Each number is the sum of the two numbers above in either the left diagonal or the right diagonal. The first few rows are:

1
1 1
2 1 2
3 2 2 3
5 3 4 3 5
8 5 6 6 5 8
13 8 10 9 10 8 13
21 13 16 15 15 16 13 21
34 21 26 24 25 24 26 21 34
55 34 42 39 40 40 39 42 34 55

(See ). The recurrence relation is H(0, 0) = H(1, 0) = H(1, 1) = H(2, 1) = 1 and H(n, j) = H(n − 1, j) + H(n − 2, j) or H(n, j) = H(n − 1, j − 1) + H(n − 2, j − 2).

The entries in the triangle satisfy the identity

H(n, i) = F(i + 1) × F(n − i + 1).

Thus, the two outermost diagonals are the Fibonacci numbers, while the numbers on the middle vertical line are the squares of the Fibonacci numbers. All the other numbers in the triangle are the product of two distinct Fibonacci numbers greater than 1. The row sums are the convolved Fibonacci numbers .