Hosoya’s triangle

Hosoya’s triangle or the Fibonacci triangle is a triangular arrangement of numbers (like Pascal’s triangle) based on the Fibonacci numbersMathworldPlanetmath. Each number is the sum of the two numbers above in either the left diagonal or the right diagonal. The first few rows are:


(See sequenceMathworldPlanetmath 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 productPlanetmathPlanetmath of two distinct Fibonacci numbers greater than 1. The row sums are the convolved Fibonacci numbers (A001629 in Sloane’s OEIS).


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