The Hosoya triangle is a construction that is similar to Pas

The Hosoya triangle is a construction that is similar to Pascal\'s triangle. Let F(n, k) denote the Hosoya coefficient for row n = 0, 1, ... and column k = 0, 1, ... (see Fig. 1). The Hosoya coefficients are defined by induction, F(0, 0) = F(l, 0) = F(1, 1) = F(2, 1) = 1 and if 2

Solution

1.a)F(0,0)=F(1.0)=F(1,1)=F(2,1)-> The first recurrence relation means that cach number is the sum of the two numbers above in either the left diagonal or the right diagonal.

F(0,0)=1=>Top most 1

F(1,0)=1=>the 1 left to F(0,0) and down

F(1,1)=>1=>The right to F(0,0) and down

F(2,1)=>1=>The 1 below between F(1,0) AND F(1,1)

b)

F(n, k) = F(n-1,k) × F(n 2,k).

This means that the two outermost diagonals are the Fibonacci numbers,and on the other hand 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.