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On a family of nested recurrences. (English) Zbl 0547.10009
Let k be an integer and let $$g_ k(n)$$ be a function defined by $$g_ k(n)=n-g_ k(g_ k(n-k))$$ for integers $$n\geq 1$$ and $$g_ k(n)=0$$ for $$n\leq 0$$. The authors give a closed form for the numbers $$g_ k(n)$$ proving that $$g_ k(n)=\sum^{k-1}_{i=0}[\psi [(n+i)/k]+\psi],$$ where $$\psi =(\sqrt{5}-1)/2$$ and [x] denotes the greatest integer function. In case $$k=1$$ the following interesting result is proved: The set of positive integers n for which $$g_ 1(n)-g_ 1(n-1)\neq 0$$ is given by the set $$B_{\phi}=\{[\phi],[2\phi],...\},$$ where $$\phi =\psi +1$$.
Reviewer: P.Kiss

##### MSC:
 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11B37 Recurrences
##### Keywords:
recursive function; nested recursive definition