Regular ternary sums of generalized octagonal numbers
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Let $P_8(x)=3x^2-2x$. A polynomial of the form $a_1P_8(x_1)+a_2P_8(x_2)+\cdots+a_kP_8(x_k)$ ($a_i\in \n$) is called a $k$-ary octagonal form. An octagonal form is called regular if it represents every positive integer which is locally represented. In this paper, we prove that there are at most 15 regular ternary octagonal forms and establish the regularity of 12 forms.
Abstract
Let $P_8(x)=3x^2-2x$. A polynomial of the form $a_1P_8(x_1)+a_2P_8(x_2)+\cdots+a_kP_8(x_k)$ ($a_i\in \n$) is called a $k$-ary octagonal form. An octagonal form is called regular if it represents every positive integer which is locally represented. In this paper, we prove that there are at most 15 regular ternary octagonal forms and establish the regularity of 12 forms.