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 The number of representations of a positive integer by triangular, square and decagonal numbers Bull. Korean Math. Soc. 2019 Vol. 56, No. 5, 1143-1157 https://doi.org/10.4134/BKMS.b180914Published online September 30, 2019 Uha Isnaini, Ray Melham, Pee Choon Toh Nanyang Technological University; University of Technology, Sydney; Nanyang Technological University Abstract : Let $T_aD_b(n)$ and $T_aD'_b(n)$ denote respectively the number of representations of a positive integer $n$ by $a(x^2-x)/2 +b(4y^2-3y)$ and $a(x^2-x)/2 +b(4y^2-y)$. Similarly, let $S_aD_b(n)$ and $S_aD'_b(n)$ denote respectively the number of representations of $n$ by $ax^2 +b(4y^2-3y)$ and $ax^2 +b(4y^2-y)$. In this paper, we prove 162 formulas for these functions. Keywords : representations by binary quadratic forms MSC numbers : Primary 11E25, 11E16 Downloads: Full-text PDF   Full-text HTML