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:

\displaystyle\begin{array}[]{cccccccccccccccccc}&&&&&&&&&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&\\ &&&&&\vdots&&&&\vdots&&&&\vdots&&&&\\ \end{array}

(See sequence A058071 in Sloaen’s OEIS). 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 $H(n,j)=H(n-1,j-1)+H(n-2,j-2)$ .

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 (A001629 in Sloane’s OEIS).

References

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

Title	Hosoya’s triangle
Canonical name	HosoyasTriangle
Date of creation	2013-03-22 18:07:47
Last modified on	2013-03-22 18:07:47
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 05A10
Synonym	Fibonacci triangle
Synonym	Hosoya triangle