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We describe several occurrences of the morphism \(1 \to 121\) , \(2 \to 12221\) and the closely related morphism \(2 \to 211\) , \(1 \to 2\) (as well as simple variants) in the literature. Furthermore we prove that a sequence in the OEIS, proposed by Kimberling, is the same as a sequence independently studied by Akiyama, Brunotte, Pethő, and Steiner related to a conjecture on the periodicity of certain piecewise affine planar maps. Finally we prove conjectures of Kimberling and conjectures of Baysal in the OEIS.

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On the Morphism \(1 \to 121\) , \(2 \to 12221\)

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  • 发表日期：

    2025年

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  • Jean-Paul Allouche

摘要

We describe several occurrences of the morphism \(1 \to 121\) , \(2 \to 12221\) and the closely related morphism \(2 \to 211\) , \(1 \to 2\) (as well as simple variants) in the literature. Furthermore we prove that a sequence in the OEIS, proposed by Kimberling, is the same as a sequence independently studied by Akiyama, Brunotte, Pethő, and Steiner related to a conjecture on the periodicity of certain piecewise affine planar maps. Finally we prove conjectures of Kimberling and conjectures of Baysal in the OEIS.