The Number of Representations of a Positive Integer by Triangular, Square, and Decagonal Numbers
Bull. Korean Math. Soc.
Published online 2019 May 16
Uha Isnaini, Ray Melham, and Pee Choon Toh
National Institute of Education, Nanyang Technological University, University of Technology, Sydney
Abstract : Let $T_aD_b(n)$, $T_aD'_b(n)$, $S_aD_b(n)$, and $S_aD'_b(n)$ denote the number of representations of a positive integer $n$ by $a(x^2-x)/2+b(4y^2-3y)$, $a(x^2-x)/2+b(4y^2-y)$, $ax^2+b(4y^2-3y)$, and $ax^2+b(4y^2-y)$ respectively. In this paper, we prove 162 formulas for these functions.
Keywords : representations by binary quadratic forms; polygonal numbers
MSC numbers : 11E25
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