The Fibonacci Quarterly

Basic Definitions and Formulas

The Fibonacci numbers F(n) and the Lucas numbers L(n) satisfy
F(n+2)=F(n+1)+F(n), F(0)=0, F(1)=1;
L(n+2)=L(n+1)+L(n), L(0)=2, L(1)=1.
Also, alpha=(1+sqrt 5)/2 and beta=(1-sqrt 5)/2, so that
F(n)=(alpha^n-beta^n)/(sqrt 5) and L(n)=alpha^n+beta^n.

The Pell numbers P(n) and their associated numbers Q(n) satisfy
P(n+2)=2P(n+1)+P(n), P(0)=0, P(1)=1;
Q(n+2)=2Q(n+1)+Q(n), Q(0)=1, Q(1)=1.
If p=1+sqrt 2 and q=1-sqrt 2, then
P(n)=(p^n-q^n)/(sqrt 8) and Q(n)=(p^n+q^n)/2.
The Pell-Lucas numbers, R(n), are given by R(n)=2Q(n).

Fibonacci numbers: 0, 1, 1, 2,  3,  5,  8, 13, 21, 34,  55,  89, 144, ...
Lucas numbers:     2, 1, 3, 4,  7, 11, 18, 29, 47, 76, 123, 199, ...
Pell numbers:      0, 1, 2, 5, 12, 29, 70, 169, 408, 985, ...

Submission Instructions

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