Padovan sequence

The first few values of P(n) are The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay Dom.

Hans van der Laan: Modern Primitive.

[2] The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996.

[3] He also writes about it in one of his books, "Math Hysteria: Fun Games With Mathematics".

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.

The Perrin sequence can be obtained from the Padovan sequence by the following formula: 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 Starting with m = −1 and working backwards, we extend P(m) to negative indices: The sum of the first n terms in the Padovan sequence is 2 less than P(n + 5), i.e.

Sums of alternate terms, sums of every third term and sums of every fifth term are also related to other terms in the sequence: Sums involving products of terms in the Padovan sequence satisfy the following identities: The Padovan sequence also satisfies the identity The Padovan sequence is related to sums of binomial coefficients by the following identity: For example, for k = 12, the values for the pair (m, n) with 2m + n = 12 which give non-zero binomial coefficients are (6, 0), (5, 2) and (4, 4), and: The Padovan sequence numbers can be written in terms of powers of the roots of the equation[1] This equation has 3 roots; one real root p (known as the plastic ratio) and two complex conjugate roots q and r.[5] Given these three roots, the Padovan sequence can be expressed by a formula involving p, q and r : where a, b and c are constants.

The ratio of successive terms in the Padovan sequence approaches p, which has a value of approximately 1.324718.

The generating function of the Padovan sequence is This can be used to prove identities involving products of the Padovan sequence with geometric terms, such as: In a similar way to the Fibonacci numbers that can be generalized to a set of polynomials called the Fibonacci polynomials, the Padovan sequence numbers can be generalized to yield the Padovan polynomials.

A spiral can be formed based on connecting the corners of a set of 3-dimensional cuboids.

Successive sides of this spiral have lengths that are the Padovan numbers multiplied by the square root of 2.

Meru[6] observed certain diagonals in Pascal's triangle (see diagram) and drew them on paper in 1993.