Nested recursions, simultaneous parameters and linear combinations of trees

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In my research I focus on developing methodologies to solve nested (also called self-referencing or meta-Fibonacci) recursions. Loosely speaking, these are recursions in which one or more of the arguments in the recursion depend upon earlier terms of the solution. The earliest examples of nested recursions appeared about forty years ago, and include the so-called ``Golomb" recursion a(n) = a(n-a(n-1)) + 1, the Hofstadter Q-recursion Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2)) and the Conolly recursion C(n) = C(n-C(n-1))+C(n-1-C(n-2)).