A Fast-Growing Sequence Inspired by TREE(k)

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Acknowledgments

The author wishes to thank Prof. Bodgan Suceava (Cal State Fullerton) for encouraging him to submit his math ideas to the College Math Journal. He also thanks Dr. David Patrick (AOPS developer) and Prof. Dongsheng Wu (University of Alabama) for their suggestions and editorial comments. He would also like to thank his school teacher Shelley Godett for her continuous support and encouragement to take challenges. Lastly, he thanks the editor and reviewers for their valuable comments and tremendous help to improve the quality of this paper.

Summary

Inspired by the tree sequence, a novel sequence of tuples is constructed. The maximum length of such a sequence is studied and it turns out the sequence grows very fast. It is remarkable to see how our results can be expressed using Knuth’s up-arrow notation and the recursive Ackermann function, and that it is associated with Eulerian numbers.

Additional information

Notes on contributors

Kevin Y. Du

Kevin Y. Du ([email protected], MR ID 1339570, ORCID 0000-0002-8280-4904) is a junior at Portola High School (Irvine, CA). He loves exploring math of many kinds. He enjoys watching YouTube math videos during his spare time (he is a big fan of 3Blue1Brown, Mathologer, Numberphile, and Vsauce). Other than math, he also enjoys coding games as well as playing saxophone in his Golden Wind Band.