Padovan sequence

 ( Integer Sequences ) 

In number theory, the Padovan sequence is the integer sequence P( n) defined. by the initial values:

$$P(0)\; =\; P(1)\; =\; P(2)\; =\; 1,$$

and the recurrence relation

$$P(n)\; =\; P(n-2)+P(n-3).$$

The first few values of P( n) are

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ...

The Padovan sequence is named after Richard Padovan who attributed its discovery to Netherlands architect Hans van der Laan in his 1994 essay Dom. Hans van der Laan: Modern Primitive.Richard Padovan. Dom Hans van der Laan: modern primitive: Architectura & Natura Press, . The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996.Ian Stewart, Tales of a Neglected Number, Scientific American, No. 6, June 1996, pp. 92-93. He also writes about it in one of his books, "Math Hysteria: Fun Games With Mathematics". .

The above definition is the one given by Ian Stewart and by MathWorld. Other sources may start the sequence at a different place, in which case some of the identities in this article must be adjusted with appropriate offsets.

---

Recurrence relations In the spiral, each triangle shares a side with two others giving a visual proof that the Padovan sequence also satisfies the recurrence relation

$P(n)=P(n-1)+P(n-5)$

Starting from this, the defining recurrence and other recurrences as they are discovered, one can create an infinite number of further recurrences by repeatedly replacing $P(m)$ by $P(m\; -\; 2)\; +\; P(m\; -\; 3)$

The Perrin sequence satisfies the same recurrence relations as the Padovan sequence, although it has different initial values.

The Perrin sequence can be obtained from the Padovan sequence by the following formula:

$\backslash mathrm\{Perrin\}(n)=P(n+1)+P(n-10).\backslash ,$

---

Extension to negative parameters As with any sequence defined by a recurrence relation, Padovan numbers P( m) for m<0 can be defined by rewriting the recurrence relation as

$P(m)\; =\; P(m+3)\; -\; P(m+1),$

Starting with m = −1 and working backwards, we extend P( m) to negative indices:

>

P−20

P−19

P−18

P−17

P−16

P−15

P−14

P−13

P−12

P−11

P−10

P−9

P−8

P−7

P−6

P−5

P−4

P−3

P−2

P−1

P0

P1

P2

7

−7

4

0

−3

4

−3

1

1

−2

2

−1

0

1

−1

1

0

0

1

0

1

1

1

---

Sums of terms

The sum of the first n terms in the Padovan sequence is 2 less than P( n + 5), i.e.

$\backslash sum\_\{m=0\}^n\; P(m)=P(n+5)-2.$

Sums of alternate terms, sums of every third term and sums of every fifth term are also related to other terms in the sequence:

$\backslash sum\_\{m=0\}^n\; P(2m)=P(2n+3)-1$

$\backslash sum\_\{m=0\}^n\; P(2m+1)=P(2n+4)-1$

$\backslash sum\_\{m=0\}^n\; P(3m)=P(3n+2)$

$\backslash sum\_\{m=0\}^n\; P(3m+1)=P(3n+3)-1$

$\backslash sum\_\{m=0\}^n\; P(3m+2)=P(3n+4)-1$

$\backslash sum\_\{m=0\}^n\; P(5m)=P(5n+1).$

Sums involving products of terms in the Padovan sequence satisfy the following identities:

$\backslash sum\_\{m=0\}^n\; P(m)^2=P(n+2)^2-P(n-1)^2-P(n-3)^2$

$\backslash sum\_\{m=0\}^n\; P(m)^2P(m+1)=P(n)P(n+1)P(n+2)$

$\backslash sum\_\{m=0\}^n\; P(m)P(m+2)=P(n+2)P(n+3)-1.$

---

Other identities

The Padovan sequence also satisfies the identity

$P(n)^2-P(n+1)P(n-1)=P(-n-7).\backslash ,$

The Padovan sequence is related to sums of binomial coefficients by the following identity:

$P(k-2)\; =\; \backslash sum\_\{2m+n=k\}\{m\; \backslash choose\; n\}\; =\; \backslash sum\_\{m=\backslash lceil\; k/3\backslash rceil\}^\{\backslash lfloor\; k/2\backslash rfloor\}\{m\; \backslash choose\; k-2m\}.$

For example, for k = 12, the values for the pair ( m,  n) with 2 m +  n = 12 which give non-zero binomial coefficients are (6, 0), (5, 2) and (4, 4), and:

$\{6\; \backslash choose\; 0\}+\{5\; \backslash choose\; 2\}+\{4\; \backslash choose\; 4\}\; =\; 1+10+1=12\; =\; P(10).\backslash ,$

---

Binet-like formula

The Padovan sequence numbers can be written in terms of powers of the roots of the equation

$x^3\; -x\; -1\; =\; 0.\backslash ,$

This equation has 3 roots; one real number root p (known as the plastic ratio) and two complex conjugate roots q and r.Richard Padovan, "Dom Hans Van Der Laan and the Plastic Number" , pp. 181-193 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002. Given these three roots, the Padovan sequence can be expressed by a formula involving p, q and r :

$P(n)\; =\; a\; p^n\; +\; b\; q^n\; +\; c\; r^n$

where a, b and c are constants.

Since the of the complex number roots q and r are both less than 1 (and hence p is a Pisot–Vijayaraghavan number), the powers of these roots approach 0 for large n, and $P(n)\; -\; a\; p^n$ tends to zero.

For all $n\; \backslash ge\; 0$, P( n) is the nearest integer to $\backslash frac\{p^5\}\{2p+3\}\; p^n$. Indeed, $\backslash frac\{p^5\}\{2p+3\}$ is the value of constant a above, while b and c are obtained by replacing p with q and r, respectively.

The ratio of successive terms in the Padovan sequence approaches p, which has a value of approximately 1.324718. This constant bears the same relationship to the Padovan sequence and the Perrin sequence as the golden ratio does to the Fibonacci sequence.

---

Combinatorial interpretations

• P( n) is the number of ways of writing n + 2 as an ordered sum in which each term is either 2 or 3 (i.e. the number of compositions of n + 2 in which each term is either 2 or 3). For example, P(6) = 4, and there are 4 ways to write 8 as an ordered sum of 2s and 3s:

:2 + 2 + 2 + 2 ; 2 + 3 + 3 ; 3 + 2 + 3 ; 3 + 3 + 2

• The number of ways of writing n as an ordered sum in which no term is 2 is P(2 n − 2). For example, P(6) = 4, and there are 4 ways to write 4 as an ordered sum in which no term is 2:

:4 ; 1 + 3 ; 3 + 1 ; 1 + 1 + 1 + 1

• The number of ways of writing n as a palindromic ordered sum in which no term is 2 is P( n). For example, P(6) = 4, and there are 4 ways to write 6 as a palindromic ordered sum in which no term is 2:

:6 ; 3 + 3 ; 1 + 4 + 1 ; 1 + 1 + 1 + 1 + 1 + 1

• The number of ways of writing n as an ordered sum in which each term is odd and greater than 1 is equal to P( n − 5). For example, P(6) = 4, and there are 4 ways to write 11 as an ordered sum in which each term is odd and greater than 1:

:11 ; 5 + 3 + 3 ; 3 + 5 + 3 ; 3 + 3 + 5

• The number of ways of writing n as an ordered sum in which each term is congruent to 2 mod 3 is equal to P( n − 4). For example, P(6) = 4, and there are 4 ways to write 10 as an ordered sum in which each term is congruent to 2 mod 3:

:8 + 2 ; 2 + 8 ; 5 + 5 ; 2 + 2 + 2 + 2 + 2

---

Generating function

The generating function of the Padovan sequence is

$G(P(n);x)=\backslash frac\{1+x\}\{1-x^2-x^3\}.$

This can be used to prove identities involving products of the Padovan sequence with geometric series, such as:

$\backslash sum\_\{n=0\}^\{\backslash infty\}\backslash frac\{P(n)\}\{2^n\}\; =\; \backslash frac\{12\}\{5\}.$

$\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{P(n)\}\{\backslash alpha^n\}\; =\; \backslash frac\{\backslash alpha^2(\backslash alpha+1)\}\{\backslash alpha^3-\backslash alpha-1\}.$

---

Generalizations

In a similar way to the that can be generalized to a set of called the Fibonacci polynomials, the Padovan sequence numbers can be generalized to yield the Padovan polynomials.

---

Padovan L-system

If we define the following simple grammar:

variables : A B C

constants : none

start : A

rules : (A → B), (B → C), (C → AB)

then this Lindenmayer system or L-system produces the following sequence of strings:

n = 0 : A

n = 1 : B

n = 2 : C

n = 3 : AB

n = 4 : BC

n = 5 : CAB

n = 6 : ABBC

n = 7 : BCCAB

n = 8 : CABABBC

and if we count the length of each string, we obtain the Padovan numbers:

1, 1, 1, 2, 2, 3, 4, 5, ...

Also, if you count the number of As, Bs and Cs in each string, then for the nth string, you have P( n − 5) As, P( n − 3) Bs and P( n − 4) Cs. The count of BB pairs and CC pairs are also Padovan numbers.

---

Cuboid spiral

A spiral can be formed based on connecting the corners of a set of 3-dimensional . This is the Padovan cuboid spiral. Successive sides of this spiral have lengths that are the Padovan numbers multiplied by the square root of 2.

---

Pascal's triangle

Erv Wilson in his paper The Scales of Mt. MeruErv Wilson (1993), Scales of Mt. Meru observed certain diagonals in Pascal's triangle (see diagram) and drew them on paper in 1993. The Padovan numbers were discovered in 1994. Paul Barry (2004) observed that these diagonals generate the Padovan sequence by summing the diagonal numbers. See formula credited to Paul Barry, July 6, 2004

• Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275, No. 5, November 1996, Pg. 118.

---

External links

• A Padovan sequence calculator

Categories: Integer Sequences, Recurrence Relations

Post Comment