A slow-growing sequence defined by an unusual recurrence

van de Bult, Fokko J.; Gijswijt, Dion C.; Linderman, John P.; Sloane, N.J.A.; Wilks, Allan R.

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A slow-growing sequence defined by an unusual recurrence

van de Bult, Fokko J. ; Gijswijt, Dion C. ; Linderman, John P. ; Sloane, N.J.A. ; Wilks, Allan R.

Journal of integer sequences, Tome 10 (2007) no. 1.

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Résumé

Summary: The sequence starts with $a(1) =1$; to extend it one writes the sequence so far as $XY^k$, where $X$ and $Y$ are strings of integers, $Y$ is nonempty and $k$ is as large as possible: then the next term is $k$. The sequence begins 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, $\ldots$ A 4 appears for the first time at position 220, but a 5 does not appear until about position $10^{10^{23}}$. The main result of the paper is a proof that the sequence is unbounded. We also present results from extensive numerical investigations of the sequence and of certain derived sequences, culminating with a heuristic argument that $t (for t=5,6, \ldots)$ appears for the first time at about position $2\uparrow (2\uparrow (3\uparrow (4\uparrow (5\uparrow \ldots \uparrow ({(t-2)}\uparrow {(t-1)})))))$, where $\uparrow$ denotes exponentiation. The final section discusses generalizations.

Classification : 11B37
Keywords: recurrence, friedman sequence, ackermann function

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     title = {A slow-growing sequence defined by an unusual recurrence},
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van de Bult, Fokko J.; Gijswijt, Dion C.; Linderman, John P.; Sloane, N.J.A.; Wilks, Allan R. A slow-growing sequence defined by an unusual recurrence. Journal of integer sequences, Tome 10 (2007) no. 1. http://geodesic.mathdoc.fr/item/JIS_2007__10_1_a4/