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ON THE PARABOLIC PARTITIONS OF A NUMBER
Keywords:
partition, parabola, arithmetic generated by a sequence.DOI:
https://doi.org/10.17654/0972555523015Abstract
The paper solves the enumeration of the set $P P(n)$ of partitions of $n \in \mathbb{N}$ in which the nondecreasing sequence of parts $p(1), p(2), \ldots$, $p(d)$ is contained in a degree-2 polynomial $p(x) \in \mathbb{Q}[x]$. This is a generalization of the partitions of a number into arithmetic progressions. We also study the problem of dividing $n$ into parts whose differences between consecutive parts are consecutive integers. In particular, we focus on the problem of the sum of consecutive triangular numbers.
Recieved: May 6, 2023
Accepted: July 1, 2023
References
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F. Javier de Vega, A complete solution of the partition of a number into arithmetic progressions, JP Journal of Algebra Number Theory and Applications 53(2) (2022), 109-122.
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