Hosoya's triangle

Hosoya's triangle or the 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

(See OEIS|id=A058071). The recurrence relation is "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).

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 convolved Fibonacci numbers (OEIS2C|id=A001629).

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