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 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 Full-Text :