Hosoya's triangle or the Hosoya triangle (originally Fibonacci triangle; ) is a triangular arrangement of numbers (like

Pascal's triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...

) based on the

Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...

s. Each number is the sum of the two numbers above in either the left diagonal or the right diagonal.

Name

The name "Fibonacci triangle" has also been used for triangles composed of Fibonacci numbers or related numbers or triangles with Fibonacci sides and integral area, hence is ambiguous.

Recurrence

The numbers in this triangle obey the

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

s :

H(0,0)=H(1,0)=H(1,1)=H(2,1)=1

and :

\begin 
H(n,j)&=H(n-1,j)+H(n-2,j)\\
&=H(n-1,j-1)+H(n-2,j-2).
\end

Relation to Fibonacci numbers

The entries in the triangle satisfy the identity :

H(n,i)=F(i+1)\cdot 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 first convolved Fibonacci numbers.

References

{{reflist Triangles of numbers Fibonacci numbers