Pell number

In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins ⁠1/1⁠, ⁠3/2⁠, ⁠7/5⁠, ⁠17/12⁠, and ⁠41/29⁠, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. 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.

Both the Pell numbers and the companion Pell numbers may be calculated by means of a recurrence relation similar to that for the Fibonacci numbers, and both sequences of numbers grow exponentially, proportionally to powers of the silver ratio 1 + √2. As well as being used to approximate the square root of two, Pell numbers can be used to find square triangular numbers, to construct integer approximations to the right isosceles triangle, and to solve certain combinatorial enumeration problems.^[1]

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.

The Pell numbers are defined by the recurrence relation:

${\displaystyle P_{n}={\begin{cases}0&{\mbox{if }}n=0;\\1&{\mbox{if }}n=1;\\2P_{n-1}+P_{n-2}&{\mbox{otherwise.}}\end{cases}}}$

In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number, plus the Pell number before that. The first few terms of the sequence are

0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, … (sequence A000129 in the OEIS).

Analogously to the Binet formula, the Pell numbers can also be expressed by the closed form formula

${\displaystyle P_{n}={\frac {\left(1+{\sqrt {2}}\right)^{n}-\left(1-{\sqrt {2}}\right)^{n}}{2{\sqrt {2}}}}.}$ For large values of n, the (1 + √2)ⁿ term dominates this expression, so the Pell numbers are approximately proportional to powers of the silver ratio 1 + √2, analogous to the growth rate of Fibonacci numbers as powers of the golden ratio.

A third definition is possible, from the matrix formula

${\displaystyle {\begin{pmatrix}P_{n+1}&P_{n}\\P_{n}&P_{n-1}\end{pmatrix}}={\begin{pmatrix}2&1\\1&0\end{pmatrix}}^{n}.}$

Many identities can be derived or proven from these definitions; for instance an identity analogous to Cassini's identity for Fibonacci numbers,

${\displaystyle P_{n+1}P_{n-1}-P_{n}^{2}=(-1)^{n},}$

is an immediate consequence of the matrix formula (found by considering the determinants of the matrices on the left and right sides of the matrix formula).^[2]

Pell numbers arise historically and most notably in the rational approximation to √2. If two large integers x and y form a solution to the Pell equation

${\displaystyle x^{2}-2y^{2}=\pm 1,}$

then their ratio ⁠x/y⁠ provides a close approximation to √2. The sequence of approximations of this form is

${\displaystyle {\frac {1}{1}},{\frac {3}{2}},{\frac {7}{5}},{\frac {17}{12}},{\frac {41}{29}},{\frac {99}{70}},\dots }$

where the denominator of each fraction is a Pell number and the numerator is the sum of a Pell number and its predecessor in the sequence. That is, the solutions have the form

${\displaystyle {\frac {P_{n-1}+P_{n}}{P_{n}}}.}$

The approximation

${\displaystyle {\sqrt {2}}\approx {\frac {577}{408}}}$

of this type was known to Indian mathematicians in the third or fourth century BCE.^[3] The Greek mathematicians of the fifth century BCE also knew of this sequence of approximations:^[4] Plato refers to the numerators as rational diameters.^[5] In the second century CE Theon of Smyrna used the term the side and diameter numbers to describe the denominators and numerators of this sequence.^[6]

These approximations can be derived from the continued fraction expansion of ${\displaystyle {\sqrt {2}}}$:

${\displaystyle {\sqrt {2}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots \,}}}}}}}}}}.}$

Truncating this expansion to any number of terms produces one of the Pell-number-based approximations in this sequence; for instance,

${\displaystyle {\frac {577}{408}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2}}}}}}}}}}}}}}.}$

As Knuth (1994) describes, the fact that Pell numbers approximate √2 allows them to be used for accurate rational approximations to a regular octagon with vertex coordinates (± P_i, ± P_i +1) and (± P_i +1, ± P_i ). All vertices are equally distant from the origin, and form nearly uniform angles around the origin. Alternatively, the points ${\displaystyle (\pm (P_{i}+P_{i-1}),0)}$, ${\displaystyle (0,\pm (P_{i}+P_{i-1}))}$, and ${\displaystyle (\pm P_{i},\pm P_{i})}$ form approximate octagons in which the vertices are nearly equally distant from the origin and form uniform angles.

A Pell prime is a Pell number that is prime. The first few Pell primes are

2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, ... (sequence A086383 in the OEIS).

The indices of these primes within the sequence of all Pell numbers are

2, 3, 5, 11, 13, 29, 41, 53, 59, 89, 97, 101, 167, 181, 191, 523, 929, 1217, 1301, 1361, 2087, 2273, 2393, 8093, ... (sequence A096650 in the OEIS)

These indices are all themselves prime. As with the Fibonacci numbers, a Pell number P_n can only be prime if n itself is prime, because if d is a divisor of n then P_d is a divisor of P_n.

The only Pell numbers that are squares, cubes, or any higher power of an integer are 0, 1, and 169 = 13².^[7]

However, despite having so few squares or other powers, Pell numbers have a close connection to square triangular numbers.^[8] Specifically, these numbers arise from the following identity of Pell numbers:

${\displaystyle {\bigl (}\left(P_{k-1}+P_{k}\right)\cdot P_{k}{\bigr )}^{2}={\frac {\left(P_{k-1}+P_{k}\right)^{2}\cdot \left(\left(P_{k-1}+P_{k}\right)^{2}-(-1)^{k}\right)}{2}}.}$

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 P_4n +1 is always a square:

${\displaystyle \sum _{i=0}^{4n+1}P_{i}=\left(\sum _{r=0}^{n}2^{r}{2n+1 \choose 2r}\right)^{\!2}=\left(P_{2n}+P_{2n+1}\right)^{2}.}$

For instance, the sum of the Pell numbers up to P₅, 0 + 1 + 2 + 5 + 12 + 29 = 49, is the square of P₂ + P₃ = 2 + 5 = 7. The numbers P_2n + P_2n +1 forming the square roots of these sums,

1, 7, 41, 239, 1393, 8119, 47321, … (sequence A002315 in the OEIS),

are known as the Newman–Shanks–Williams (NSW) numbers.

If a right triangle has integer side lengths a, b, c (necessarily satisfying the Pythagorean theorem a² + b² = c²), 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

${\displaystyle \left(2P_{n}P_{n+1},P_{n+1}^{2}-P_{n}^{2},P_{n+1}^{2}+P_{n}^{2}=P_{2n+1}\right).}$

The sequence of Pythagorean triples formed in this way is

(4,3,5), (20,21,29), (120,119,169), (696,697,985), …

The companion Pell numbers or Pell–Lucas numbers are defined by the recurrence relation

${\displaystyle Q_{n}={\begin{cases}2&{\mbox{if }}n=0;\\2&{\mbox{if }}n=1;\\2Q_{n-1}+Q_{n-2}&{\mbox{otherwise.}}\end{cases}}}$

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 82 = 2 × 34 + 14 = 70 + 12. The first few terms of the sequence are (sequence A002203 in the OEIS): 2, 2, 6, 14, 34, 82, 198, 478, …

Like the relationship between Fibonacci numbers and Lucas numbers,

${\displaystyle Q_{n}={\frac {P_{2n}}{P_{n}}}}$

for all natural numbers n.

The companion Pell numbers can be expressed by the closed form formula

${\displaystyle Q_{n}=\left(1+{\sqrt {2}}\right)^{n}+\left(1-{\sqrt {2}}\right)^{n}.}$

These numbers are all even; each such number is twice the numerator in one of the rational approximations to ${\displaystyle {\sqrt {2}}}$ discussed above.

Like the Lucas sequence, if a Pell–Lucas number ⁠1/2⁠Q_n 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, … (sequence A086395 in the OEIS).

For these n are

2, 3, 4, 5, 7, 8, 16, 19, 29, 47, 59, 163, 257, 421, … (sequence A099088 in the OEIS).

The following table gives the first few powers of the silver ratio δ = δ_S = 1 + √2 and its conjugate δ = 1 − √2.

n	(1 + √2)n	(1 − √2)n
0	1 + 0√2 = 1	1 − 0√2 = 1
1	1 + 1√2 = 2.41421…	1 − 1√2 = −0.41421…
2	3 + 2√2 = 5.82842…	3 − 2√2 = 0.17157…
3	7 + 5√2 = 14.07106…	7 − 5√2 = −0.07106…
4	17 + 12√2 = 33.97056…	17 − 12√2 = 0.02943…
5	41 + 29√2 = 82.01219…	41 − 29√2 = −0.01219…
6	99 + 70√2 = 197.9949…	99 − 70√2 = 0.0050…
7	239 + 169√2 = 478.00209…	239 − 169√2 = −0.00209…
8	577 + 408√2 = 1153.99913…	577 − 408√2 = 0.00086…
9	1393 + 985√2 = 2786.00035…	1393 − 985√2 = −0.00035…
10	3363 + 2378√2 = 6725.99985…	3363 − 2378√2 = 0.00014…
11	8119 + 5741√2 = 16238.00006…	8119 − 5741√2 = −0.00006…
12	19601 + 13860√2 = 39201.99997…	19601 − 13860√2 = 0.00002…

The coefficients are the half-companion Pell numbers H_n and the Pell numbers P_n which are the (non-negative) solutions to H² − 2P² = ±1. A square triangular number is a number

${\displaystyle N={\frac {t(t+1)}{2}}=s^{2},}$

which is both the t-th triangular number and the s-th square number. A near-isosceles Pythagorean triple is an integer solution to a² + b² = c² where a + 1 = b.

The next table shows that splitting the odd number H_n into nearly equal halves gives a square triangular number when n is even and a near isosceles Pythagorean triple when n is odd. All solutions arise in this manner.

n	Hn	Pn	t	t + 1	s	a	b	c
0	1	0	0	1	0
1	1	1				0	1	1
2	3	2	1	2	1
3	7	5				3	4	5
4	17	12	8	9	6
5	41	29				20	21	29
6	99	70	49	50	35
7	239	169				119	120	169
8	577	408	288	289	204
9	1393	985				696	697	985
10	3363	2378	1681	1682	1189
11	8119	5741				4059	4060	5741
12	19601	13860	9800	9801	6930

The half-companion Pell numbers H_n and the Pell numbers P_n can be derived in a number of easily equivalent ways.

${\displaystyle \left(1+{\sqrt {2}}\right)^{n}=H_{n}+P_{n}{\sqrt {2}}}$

${\displaystyle \left(1-{\sqrt {2}}\right)^{n}=H_{n}-P_{n}{\sqrt {2}}.}$

From this it follows that there are closed forms:

${\displaystyle H_{n}={\frac {\left(1+{\sqrt {2}}\right)^{n}+\left(1-{\sqrt {2}}\right)^{n}}{2}}.}$

and

${\displaystyle P_{n}{\sqrt {2}}={\frac {\left(1+{\sqrt {2}}\right)^{n}-\left(1-{\sqrt {2}}\right)^{n}}{2}}.}$

${\displaystyle H_{n}={\begin{cases}1&{\mbox{if }}n=0;\\H_{n-1}+2P_{n-1}&{\mbox{otherwise.}}\end{cases}}}$

${\displaystyle P_{n}={\begin{cases}0&{\mbox{if }}n=0;\\H_{n-1}+P_{n-1}&{\mbox{otherwise.}}\end{cases}}}$

Let n be at least 2.

${\displaystyle H_{n}=(3P_{n}-P_{n-2})/2=3P_{n-1}+P_{n-2};}$

${\displaystyle P_{n}=(3H_{n}-H_{n-2})/4=(3H_{n-1}+H_{n-2})/2.}$

${\displaystyle {\begin{pmatrix}H_{n}\\P_{n}\end{pmatrix}}={\begin{pmatrix}1&2\\1&1\end{pmatrix}}{\begin{pmatrix}H_{n-1}\\P_{n-1}\end{pmatrix}}={\begin{pmatrix}1&2\\1&1\end{pmatrix}}^{n}{\begin{pmatrix}1\\0\end{pmatrix}}.}$

So

${\displaystyle {\begin{pmatrix}H_{n}&2P_{n}\\P_{n}&H_{n}\end{pmatrix}}={\begin{pmatrix}1&2\\1&1\end{pmatrix}}^{n}.}$

The difference between H_n and P_n√2 is

${\displaystyle \left(1-{\sqrt {2}}\right)^{n}\approx (-0.41421)^{n},}$

which goes rapidly to zero. So

${\displaystyle \left(1+{\sqrt {2}}\right)^{n}=H_{n}+P_{n}{\sqrt {2}}}$

is extremely close to 2H_n.

From this last observation it follows that the integer ratios ⁠H_n/P_n⁠ rapidly approach √2; and ⁠H_n/H_n −1⁠ and ⁠P_n/P_n −1⁠ rapidly approach 1 + √2.

Since √2 is irrational, we cannot have ⁠H/P⁠ = √2, i.e.,

${\displaystyle {\frac {H^{2}}{P^{2}}}={\frac {2P^{2}}{P^{2}}}.}$

The best we can achieve is either

${\displaystyle {\frac {H^{2}}{P^{2}}}={\frac {2P^{2}-1}{P^{2}}}\quad {\mbox{or}}\quad {\frac {H^{2}}{P^{2}}}={\frac {2P^{2}+1}{P^{2}}}.}$

The (non-negative) solutions to H² − 2P² = 1 are exactly the pairs (H_n, P_n) with n even, and the solutions to H² − 2P² = −1 are exactly the pairs (H_n, P_n) with n odd. To see this, note first that

${\displaystyle H_{n+1}^{2}-2P_{n+1}^{2}=\left(H_{n}+2P_{n}\right)^{2}-2\left(H_{n}+P_{n}\right)^{2}=-\left(H_{n}^{2}-2P_{n}^{2}\right),}$

so that these differences, starting with H²
₀ − 2P²
₀ = 1, are alternately 1 and −1. Then note that every positive solution comes in this way from a solution with smaller integers since

${\displaystyle (2P-H)^{2}-2(H-P)^{2}=-\left(H^{2}-2P^{2}\right).}$

The smaller solution also has positive integers, with the one exception: H = P = 1 which comes from H₀ = 1 and P₀ = 0.

The required equation

${\displaystyle {\frac {t(t+1)}{2}}=s^{2}}$

is equivalent to ${\displaystyle 4t^{2}+4t+1=8s^{2}+1,}$ which becomes H² = 2P² + 1 with the substitutions H = 2t + 1 and P = 2s. Hence the n-th solution is

${\displaystyle t_{n}={\frac {H_{2n}-1}{2}}\quad {\mbox{and}}\quad s_{n}={\frac {P_{2n}}{2}}.}$

Observe that t and t + 1 are relatively prime, so that ⁠t (t + 1)/2⁠ = s² happens exactly when they are adjacent integers, one a square H² and the other twice a square 2P². Since we know all solutions of that equation, we also have

${\displaystyle t_{n}={\begin{cases}2P_{n}^{2}&{\mbox{if }}n{\mbox{ is even}};\\H_{n}^{2}&{\mbox{if }}n{\mbox{ is odd.}}\end{cases}}}$

and ${\displaystyle s_{n}=H_{n}P_{n}.}$

This alternate expression is seen in the next table.

n	Hn	Pn	t	t + 1	s	a	b	c
0	1	0
1	1	1	1	2	1	3	4	5
2	3	2	8	9	6	20	21	29
3	7	5	49	50	35	119	120	169
4	17	12	288	289	204	696	697	985
5	41	29	1681	1682	1189	4059	4060	5741
6	99	70	9800	9801	6930	23660	23661	33461

The equality c² = a² + (a + 1)² = 2a² + 2a + 1 occurs exactly when 2c² = 4a² + 4a + 2 which becomes 2P² = H² + 1 with the substitutions H = 2a + 1 and P = c. Hence the n-th solution is a_n = ⁠H_2n +1 − 1/2⁠ and c_n = P_2n +1.

The table above shows that, in one order or the other, a_n and b_n = a_n + 1 are H_n H_n +1 and 2P_n P_n +1 while c_n = H_n +1 P_n + P_n +1 H_n.

• Weisstein, Eric W. "Pell Number". MathWorld.
• OEIS sequence A001333 (Numerators of continued fraction convergents to sqrt(2))—The numerators of the same sequence of approximations