So I know the Fibonacci sequence is defined recursively by: F(n) = F(n-1) + F(n-2). I also know that it has a closed-form definition, Binet's formula.
But what about sequences such as:
F(n) = F(n-1) + F(n-2) + F(n-3)
F(n) = F(n-1) + F(n-2) + F(n-3) + F(n-4)
...
and so on.
Do those have names and/or already-known closed-form solutions? Better yet, would it be possible to derive a general solution for all of the fibonacci-like sequences I'm describing? Such that Binet's formula is a special case of the general formula?
But what about sequences such as:
F(n) = F(n-1) + F(n-2) + F(n-3)
F(n) = F(n-1) + F(n-2) + F(n-3) + F(n-4)
...
and so on.
Do those have names and/or already-known closed-form solutions? Better yet, would it be possible to derive a general solution for all of the fibonacci-like sequences I'm describing? Such that Binet's formula is a special case of the general formula?
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http://en.wikipedia.org/wiki/Generalizat…
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Even so, the wikipedia article on it is surprisingly good, in spite of all the criticism that wikipedia gets for "shabby research".
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