Hosoya's triangle

The Hosoya triangle or Hosoya's triangle (originally Fibonacci 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)).

Name

The name "Fibonacci triangle" has also been used for triangles composed of Fibonacci numbers or related numbers—Wilson (1998), or triangles with Fibonacci sides and integral area—Yuan (1999), hence is ambiguous.

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)

= 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 (1976), "Fibonacci Triangle", The Fibonacci Quarterly, vol. 14, no. 2, pp. 173–178.
• Thomas Koshy (2001), Fibonacci and Lucas Numbers and Applications, pp. 187–195. New York: Wiley.
• Brad Wilson (1998), "The Fibonacci triangle modulo p". The Fibonacci Quarterly, vol. 36, no. 3, pp. 194–203.
• Ming Hao Yuan (1999), "A result on a conjecture concerning the Fibonacci triangle when k=4." (In Chinese.) Journal of Huanggang Normal University, vol. 19, no. 4, pp. 19–23.