Generalisation of Fibonacci

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Somehow a generalisation of the fibonacci numbers, do numbers created by the formula $ F(n) = F(n-1) + [F(n-1)-F(n-2)+F(n-3)-F(n-4)+F(n-5)-F(n-6).....]$ with $F(1) = 1$ have a specific name?

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You need more starting terms than just one unless you define $F(0)$ and negative indices zero. If you did not intend $F(n-1)$ to be on the right twice, there are some values in Wikipedia, which has links into OEIS, this one for six on the right