Pell number

Specifically, these numbers arise from the following identity of Pell numbers:

Besides 1 and 70, no generalized heptagonal numbers are also Pell numbers.

Pell numbers arise historically and most notably in the rational approximation to the square root of 2.

However, despite having so few squares or other powers, Pell numbers have a close connection to square triangular numbers.

It is the ninth Fibonacci number and a companion Pell number.

As with the Fibonacci numbers, a Pell number can only be prime if n itself is prime.

This sequence of denominators is a particular Lucas sequence known as the Pell numbers.

A Pell prime is a Pell number that is prime.

Fourteen is a Companion Pell number.

Each term in this sequence is half the corresponding term in the sequence of companion Pell numbers.

There are many identities about the Pell numbers, and these translate into identities about the square triangular numbers.

Pell's equation and the Pell number are both named after 17th century mathematician John Pell.

The Ammann-Beenker tilings are closely related to the silver ratio () and the Pell numbers.

Starting with that same triplet and trading x and z before each iteration gives the triples with Pell numbers.

It contains an account of 'side and diameter numbers', the sequence of best rational approximations to the square root of 2, the denominators of which are Pell numbers.

As Knuth (1994) describes, the fact that Pell numbers approximate allows them to be used for accurate rational approximations to a regular octagon with vertex coordinates and .

In mathematics, the Pell numbers are an infinite sequence of integers that have been known since ancient times, the denominators of the closest rational approximations to the square root of 2.

Similar connections exist also between p and the Padovan sequence, between the golden ratio and Fibonacci numbers, and between the silver ratio and Pell numbers.

As Martin (1875) describes, the Pell numbers can be used to form Pythagorean triples in which a and b are one unit apart, corresponding to right triangles that are nearly isosceles.

This ancient Indian approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, that can be derived from the continued fraction expansion of .

Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Lucas numbers, Jacobsthal numbers, and a superset of Fermat numbers.

The numerators of the same sequence of approximations are half the companion Pell numbers or Pell-Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82.

As with Pell's equation, the name of the Pell numbers stems from Leonhard Euler's mistaken attribution of the equation and the numbers derived from it to John Pell.

The Pell-Lucas numbers are also named after Édouard Lucas, who studied sequences defined by recurrences of this type; the Pell and companion Pell numbers are Lucas sequences.

Santana and Diaz-Barrero (2006) prove another identity relating Pell numbers to squares and showing that the sum of the Pell numbers up to is always a square: