Fibonacci Numbers - Generalizations

Generalizations

The Fibonacci sequence has been generalized in many ways. These include:

• Generalizing the index to negative integers to produce the Negafibonacci numbers.
• Generalizing the index to real numbers using a modification of Binet's formula.
• Starting with other integers. Lucas numbers have L₁ = 1, L₂ = 3, and L_n = L_n−1 + L_n−2. Primefree sequences use the Fibonacci recursion with other starting points in order to generate sequences in which all numbers are composite.
• Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have P_n = 2P_{n – 1} + P_{n – 2}.
• Not adding the immediately preceding numbers. The Padovan sequence and Perrin numbers have P(n) = P(n – 2) + P(n – 3).
• Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as n-Step Fibonacci numbers.
• Adding other objects than integers, for example functions or strings—one essential example is Fibonacci polynomials.