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 an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...

) based on the

Fibonacci number In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...

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