Padovan sequence
The Padovan sequence is the sequence of integers P(n) defined by the initial values
and the recurrence relation
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. 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.
In the spiral, each triangle shares a side with two others giving a visual proof that the Padovan sequence also satisfies the recurrence relation
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 by
...
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