Journal of Integer Sequences, Vol. 25 (2022), Article 22.6.6

Proof of a Conjecture of Stanley About Stern's Array

David E. Speyer
Department of Mathematics
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530 Church Street
Ann Arbor, MI 48109-1043


Stanley, building on work of Stern, defined an array of numbers as follows: the first row of the array is ···0001000···. Each successive row is obtained by copying down the previous row and inserting, between each pair of numbers from the previous row, the sum of that pair of numbers. Let snr be the sum of the r-th powers of the elements in the n-th row of the array. Stanley showed that, for each positive integer r, the sequence snr obeys a homogeneous linear recurrence in n of length r/2 + O(1). Numerical evidence, however, suggested that snr obeys shorter recurrences, of length r/3 + O(1). We prove Stanley's conjecture.

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(Concerned with sequences A337277.)

Received March 15 2022; revised versions received June 19 2022; July 6 2022. Published in Journal of Integer Sequences, July 18 2022.

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