numbers: The left side of this identity describes a square number, while the right side describes a triangular number, so the result is a square triangular number. Falcón and Díaz-Barrero (2006) proved another identity relating Pell numbers to squares and showing that the sum of the Pell numbers up to P4n&hairsp;+1 is always a square: For instance, the sum of the Pell numbers up to P5, , is the square of . The numbers forming the square roots of these sums, 1, 7, 41, 239, 1393, 8119, 47321, … , are known as the Newman–Shanks–Williams (NSW) numbers. Pythagorean triples If a right triangle has integer side lengths a, b, c (necessarily satisfying the Pythagorean theorem ), then (a,b,c) is known as a Pythagorean triple. 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. Each such triple has the form The sequence of Pythagorean triples formed in this way is (4,3,5), (20,21,29), (120,119,169), (696,697,985), … Pell–Lucas numbers The companion Pell numbers or Pell–Lucas numbers are defined by the recurrence relation In words: the first two numbers in the sequence are both 2, and each successive number is formed by adding twice the previous Pell–Lucas number to the Pell–Lucas number before that, or equivalently, by adding the next Pell number to the previous Pell number: thus, 82 is the companion to 29, and The first few terms of the sequence are : 2, 2, 6, 14, 34, 82, 198, 478, … Like the relationship between Fibonacci numbers and Lucas numbers, for all natural numbers n. The companion Pell numbers can be expressed by the closed form formula These numbers are all even; each such number is twice the numerator in one of the rational approximations to discussed above. Like the Lucas sequence, if a Pell–Lucas number Qn is prime, it is necessary that n be either prime or a power of 2. The Pell–Lucas primes are 3, 7, 17, 41, 239, 577, … . For these n are 2, 3, 4, 5, 7, 8, 16, 19, 29, 47, 59, 163, 257, 421, … . Computations and connections The following table gives the first few powers of the silver ratio δ = δS = 1 + and its conjugate = 1 − . {| class="wikitable" style="text-align:center" |- ! n ! (1