Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A nondecreasing sequence of positive integers is $(\alpha,\beta)$-Conolly, or Conolly-like for short, if for every positive integer $m$ the number of times that $m$ occurs in the sequence is $\alpha + \beta r_m$, where $r_m$ is $1$ plus the 2-adic valuation of $m$. A recurrence relation is $(\alpha, \beta)$-Conolly if it has an $(\alpha, \beta)$-Conolly solution sequence.  discover that Conolly-like sequences often appear as solutions to nested (or meta-Fibonacci) recurrence relations of the form $A(n) = \sum_{i=1}^k A(n-s_i-\sum_{j=1}^{p_i} A(n-a_{ij}))$ with appropriate initial conditions. For any fixed integers $k$ and $p_1,p_2,\ldots, p_k$  prove that there are only finitely many pairs $(\alpha, \beta)$ for which $A(n)$ can be $(\alpha, \beta)$-Conolly. For the case where $\alpha =0$ and $\beta =1$,  provide a bijective proof using labeled infinite trees to show that, in addition to the original Conolly recurrence, the recurrence $H(n)=H(n-H(n-2)) + H(n-3-H(n-5))$ also has the Conolly sequence as a solution. When $k=2$ and $p_1=p_2$,  construct an example of an $(\alpha,\beta)$-Conolly recursion for every possible ($\alpha,\beta)$ pair, thereby providing the first examples of nested recursions with $p_i>1$ whose solutions are completely understood. Finally, in the case where $k=2$ and $p_1=p_2$,  provide an if and only if condition for a given nested recurrence $A(n)$ to be $(\alpha,0)$-Conolly by proving a very general ceiling function identity.
['Alejandro Erickson', 'Abraham Isgur', 'Bradley W. Jackson', 'Frank Ruskey', 'Stephen M. Tanny']
I am writing a research paper on a idea Nested Recurrence Relations with Conolly-like Solutions
[{"from": "human", "value": "I am writing a research paper on a idea Nested Recurrence Relations with Conolly-like Solutions"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A nondecreasing sequence of positive integers is $(\\alpha,\\beta)$-Conolly, or Conolly-like for short, if for every positive integer $m$ the number of times that $m$ occurs in the sequence is $\\alpha + \\beta r_m$, where $r_m$ is $1$ plus the 2-adic valuation of $m$. A recurrence relation is $(\\alpha, \\beta)$-Conolly if it has an $(\\alpha, \\beta)$-Conolly solution sequence.  discover that Conolly-like sequences often appear as solutions to nested (or meta-Fibonacci) recurrence relations of the form $A(n) = \\sum_{i=1}^k A(n-s_i-\\sum_{j=1}^{p_i} A(n-a_{ij}))$ with appropriate initial conditions. For any fixed integers $k$ and $p_1,p_2,\\ldots, p_k$  prove that there are only finitely many pairs $(\\alpha, \\beta)$ for which $A(n)$ can be $(\\alpha, \\beta)$-Conolly. For the case where $\\alpha =0$ and $\\beta =1$,  provide a bijective proof using labeled infinite trees to show that, in addition to the original Conolly recurrence, the recurrence $H(n)=H(n-H(n-2)) + H(n-3-H(n-5))$ also has the Conolly sequence as a solution. When $k=2$ and $p_1=p_2$,  construct an example of an $(\\alpha,\\beta)$-Conolly recursion for every possible ($\\alpha,\\beta)$ pair, thereby providing the first examples of nested recursions with $p_i>1$ whose solutions are completely understood. Finally, in the case where $k=2$ and $p_1=p_2$,  provide an if and only if condition for a given nested recurrence $A(n)$ to be $(\\alpha,0)$-Conolly by proving a very general ceiling function identity."}]