Hosoya's triangle

Fibonacci triangle or the Hosoya's triangle is a triangular arrangement of numbers (like Pascal's triangle) based on the Fibonacci numbers. 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
                 89    55    68    63    65    64    65    63    68    55    89
             144    89   110   102   105   104   104   105   102   110    89   144
                                              etc.

(See (sequence A058071 in the OEIS)).

Recurrence

The numbers in this triangle obey the recurrence relations

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).

Relation to Fibonacci numbers

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 first convolved Fibonacci numbers.

References

• Haruo Hosoya, "Fibonacci Triangle" The Fibonacci Quarterly 14 2 (1976): 173–178
• Thomas Koshy, Fibonacci and Lucas Numbers and Applications. New York: Wiley & Sons (2001): 187–195