Fibonacci Number - 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 Ln = Ln−1 + Ln−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 Pn = 2Pn – 1 + Pn – 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.

Read more about this topic:  Fibonacci Number