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单词 HosoyasTriangle
释义

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:

111212322353435856658138109108132113161515161321

(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).

References

  • 1 Haruo Hosoya, “Fibonacci Triangle” The Fibonacci Quarterly 14 2 (1976): 173 - 178
  • 2 Thomas Koshy, Fibonacci and Lucas NumbersMathworldPlanetmath and Applications. New York: Wiley & Sons (2001): 187 - 195
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更新时间:2025/5/4 6:14:28