Journal of Integer Sequences, Vol. 26 (2023), Article 23.1.2

Van der Laan Sequences and a Conjecture on Padovan Numbers

David Nacin
Department of Mathematics
William Paterson University
Wayne, NJ 07470


The Padovan sequence has the property that the largest of any four consecutive terms equals the sum of the two smallest. We examine when sequences with this property can merge with multiples of the Padovan sequence, and show that any increasing sequence with this property is a linear combination of Padovan sequences. We then show that linear combinations of Fibonacci numbers arise when we weaken the condition to sequences that increase following certain permutation patterns.

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(Concerned with sequences A000045 A000931 A177704 A291289 A321341 A321664 A327035 A328943.)

Received August 10 2022; revised version received December 27 2022. Published in Journal of Integer Sequences, December 30 2022.

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