In number theory, the Padovan sequence is the sequence of integers P(n) defined by the initial values:
P ( 0 ) = P ( 1 ) = P ( 2 ) = 1 , {\displaystyle P(0)=P(1)=P(2)=1,}
and the recurrence relation
P ( n ) = P ( n − 2 ) + P ( n − 3 ) . {\displaystyle 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, ... (sequence A000931 in the OEIS)
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. The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996. 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 ) {\displaystyle 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 ) {\displaystyle P(m)} by P ( m − 2 ) + P ( m − 3 ) {\displaystyle 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:
P e r r i n ( n ) = P ( n + 1 ) + P ( n − 10 ) . {\displaystyle \mathrm {Perrin} (n)=P(n+1)+P(n-10).\,}
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 ) , {\displaystyle P(m)=P(m+3)-P(m+1),}
Starting with m = −1 and working backwards, we extend P(m) to negative indices:
Sums of terms The sum of the first n terms in the Padovan sequence is 2 less than P(n + 5), i.e.
∑ m = 0 n P ( m ) = P ( n + 5 ) − 2. {\displaystyle \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:
∑ m = 0 n P ( 2 m ) = P ( 2 n + 3 ) − 1 {\displaystyle \sum _{m=0}^{n}P(2m)=P(2n+3)-1} OEIS: A077855
∑ m = 0 n P ( 2 m + 1 ) = P ( 2 n + 4 ) − 1 {\displaystyle \sum _{m=0}^{n}P(2m+1)=P(2n+4)-1}
∑ m = 0 n P ( 3 m ) = P ( 3 n + 2 ) {\displaystyle \sum _{m=0}^{n}P(3m)=P(3n+2)} OEIS: A034943
∑ m = 0 n P ( 3 m + 1 ) = P ( 3 n + 3 ) − 1 {\displaystyle \sum _{m=0}^{n}P(3m+1)=P(3n+3)-1}
∑ m = 0 n P ( 3 m + 2 ) = P ( 3 n + 4 ) − 1 {\displaystyle \sum _{m=0}^{n}P(3m+2)=P(3n+4)-1}
∑ m = 0 n P ( 5 m ) = P ( 5 n + 1 ) . {\displaystyle \sum _{m=0}^{n}P(5m)=P(5n+1).} OEIS: A012772 Sums involving products of terms in the Padovan sequence satisfy the following identities:
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