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Proof of a Conjecture of Stanley About Stern's Array
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David E. Speyer

Department of Mathematics

2844 East Hall

530 Church Street

Ann Arbor, MI 48109-1043

USA

**Abstract:**

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 *s*_{n}^{r} 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 *s*_{n}^{r} obeys a homogeneous linear recurrence in *n* of
length *r*/2 + O(1).
Numerical evidence, however, suggested that *s*_{n}^{r} 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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