On families of solutions for meta-fibonacci recursions related to hofstadter-conway $10000 sequence

Abstract

Hofstadter-Conway $10000 sequence is a famous meta-Fibonacci sequence which is defined by recurrence relation c(n) = c(c(n-1)) + c(n-c(n-1)) with c(1)=c(2)=1. Based on its amazing fractal-like structure, in this study, we investigate certain solution families of the new meta-Fibonacci recursion families which are ai(n) = n − ai(ai(n − i)) − ai(n − ai(n − i)) and ai(n) = ai(ai(n-i)) + ai(n-ai(n-1)). While we prove their essential properties, we also explore curious sensitivity of these recursions to initial condition selections in a reasonable approach.