In mathematics, the Pell numbers are an integer sequence that has been known at least since 400 BCE.[1] The sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number and the Pell number before that. The first few terms of the sequence are

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

The Half companion Pell numbers are another sequence which begins 1, 1, 3, 7, 17, 41... and has the same recurrence. Theon of Smyrna defined the two sequences by a paired recurrence and observed that the square of each number in this second sequence is one more or one less than the square of the corresponding Pell number (32=2•22+1 and 72=2•52-1 and so on). Consequently, this second sequence suplies the numerators, and the Pell numbers the denominators, of the sequence 1/1, 3/2, 7/5, 17/12, 41/29 ... of closest rational approximations to the square root of 2. The doubles of the numerators are sometimes called companion Pell numbers or Pell-Lucas numbers; these numbers form a second infinite sequence that begins with 2, 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 nearly-isosceles integer right triangles, and to solve certain combinatorial enumeration problems.[2]

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 Edouard Lucas, who studied sequences defined by recurrences of this type; the Pell and companion Pell numbers are Lucas sequences.

Pell numbers

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The Pell numbers are defined by the recurrence relation

 

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 and the Pell number before that.

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

 

For large values of n, the   term dominates this expression, so the Pell numbers are approximately proportional to powers of the silver ratio  , analogous to the growth rate of Fibonacci numbers as powers of the golden ratio.

A third definition is possible, from the matrix formula

 

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

 

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).[3]

Approximation to the square root of two

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Rational approximations to regular octagons, with coordinates derived from the Pell numbers.

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

 

then their ratio   provides a close approximation to  . The sequence of approximations of this form is

 

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  . The approximation

 

of this type was known to Indian mathematicians in the third or fourth century B.C.[4] The Greek mathematicians of the fifth century B.C. also knew of this sequence of approximations;[5] they called the denominators and numerators of this sequence side and diameter numbers and the numerators were also known as rational diagonals or rational diameters.[6]

These approximations can be derived from the continued fraction expansion of  :

 

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

 

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  . All vertices are equally distant from the origin, and form nearly uniform angles around the origin. Alternatively, the points  ,  , and   form approximate octagons in which the vertices are nearly equally distant from the origin and form uniform angles.

Primes and squares

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

2, 5, 29, 5741, ... (sequence A086383 in the OEIS).

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

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:

 

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.

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:

 

For instance, the sum of the Pell numbers up to  ,  , is the square of  . The numbers   forming the square roots of these sums,

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

are known as the NSW numbers.

Pythagorean triples

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Integer right triangles with nearly equal legs, derived from the Pell 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

 

The sequence of Pythagorean triples formed in this way is

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

Companion Pell numbers (Pell-Lucas numbers)

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

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.

Computations and Connections

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The following table gives the first few powers of the silver ratio   and its conjugate  .

     
0    
1    
2    
3    
4    
5    
6    
7    
8    
9    
10    
11    
12    

The coefficients are the Pell numbers and the Half companion Pell numbers   and The Pell numbers   which are the (non-negative) solutions to  . A Square triangular number is a number   which is both the  th triangular number and the  th square number. A near isosceles Pythagorean triple is an integer solution to   where  . $H_n$ is always odd. The next table shows that splitting $H_n$ 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.

      t t+1 s a b c
0 1 0 0 0 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

Definitions

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The half companion Pell Numbers   and the Pell numbers   can be derived in a number of easily equivalent ways:

Raising to powers:

 
 

From this it follows that there are closed forms:

 

and

 

Paired recurrences:

 
 

and Matrix formulations:

 

So

 

Approximations

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The difference between   and   is   which goes rapidly to zero. So   is extremely close  .

From this last observation it follows that the integer ratios   rapidly approach   while   and   rapidly approach  .

The Pell Equation  

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Since   is irrational, we can't have   i.e.  . The best we can achieve is either   or  .

The (non-negative) solutions to   are exactly the pairs   even and the solutions to   are exactly the pairs   odd. To see this, note first that

 

so that these differences, starting with   are alternately  . Then note that that every positive solution comes in this way from a solution with smaller integers since  . The smaller solution also has positive integers with the one exception   which comes from  .

Square triangular numbers

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The required equation   is equivalent to   which becomes   with the substitutions  . Hence the nth solution is   and  

Observe that   and   are relatively prime so that   happens exactly when they are adjacent integers, one a square  and the other twice a square  . Since we know all solutions of that equation, we also have

 

and  

This alternate expression is seen in the next table.

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

Pythagorean Triples

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The equality   occurs exactly when   which becomes   with the substitutions  . Hence the nth solution is   and  

The table above shows that in one order or the other   are   while  

Notes

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  1. ^ See Knorr (1976) for the fifth century date, which matches Proclus' claim that the side and diameter numbers were discovered by the Pythagoreans. For more detailed exploration of later Greek knowledge of these numbers see Thompson (1929), Vedova (1951), Ridenhour (1986), Knorr (1998), and Filep (1999).
  2. ^ For instance, Sellers (2002) proves that the number of perfect matchings in the Cartesian product of a path graph and the graph K4-e can be calculated as the product of a Pell number with the corresponding Fibonacci number.
  3. ^ For the matrix formula and its consequences see Ercolano (1979) and Kilic and Tasci (2005). Additional identities for the Pell numbers are listed by Horadam (1971) and Bicknell (1975).
  4. ^ As recorded in the Shulba Sutras; see e.g. Dutka (1986), who cites Thibaut (1875) for this information.
  5. ^ See Knorr (1976) for the fifth century date, which matches Proclus' claim that the side and diameter numbers were discovered by the Pythagoreans. For more detailed exploration of later Greek knowledge of these numbers see Thompson (1929), Vedova (1951), Ridenhour (1986), Knorr (1998), and Filep (1999).
  6. ^ For instance, as several of the references from the previous note observe, in Plato's Republic there is a reference to the "rational diameter of 5", by which Plato means 7, the numerator of the approximation 7/5 of which 5 is the denominator.
  7. ^ Pethő (1992); Cohn (1996). Although the Fibonacci numbers are defined by a very similar recurrence to the Pell numbers, Cohn writes that an analogous result for the Fibonacci numbers seems much more difficult to prove.
  8. ^ Sesskin (1962). See the square triangular number article for a more detailed derivation.

References

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unrelated

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