On the number of representations of n as a linear combination of four triangular numbers

Ramanujan's theta functions and linear combinations of four triangular numbers

Let $\Bbb Z$ and $\Bbb Z^+$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb Z^+$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2+dw(w+1)/2$ $(x,y,z,w\in\Bbb Z)$. In this paper, by using Ramanujan's theta functions $φ(q)$ and $ψ(q)$ we present many formulas and conjectures on $t(a,b,c,d;n)$.

On sums of triangular numbers

We study decompositions of natural numbers into triangular summands. For instance, we prove that any natural number can be represented as a sum of four triangular numbers, two of them having even indices and the other two having odd indices. This settles a conjecture by Zhi-Wei Sun (arXiv:1503.03743v8).

关于三个 theta 函数的乘积及整数表示为涉及平方数、三角数、五角数和八边形数的混合三元和的表示数

On a product of three theta functions and the number of representations of integers as mixed ternary sums involving squares, triangular, pentagonal and octagonal numbers

本文推导了三个theta函数乘积的线性组合一般公式，并由此得到两个theta函数乘积的公式，进而应用于整数表示为平方数、三角数、广义五角数和广义八边形数的混合三元和问题。

On the number of representations of n as a linear combination of four triangular numbers II

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$ $(x,y,z,w\in\Bbb Z$). In this paper we reveal the connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$. Suppose $a,n\in\Bbb N$ and $2\nmid a$. We show that $$t(a,b,c,d;n)=\frac 23N(a,b,c,d;8n+a+b+c+d)-2N(a,b,c,d;2n+(a+b+c+d)/4)$$ for $(a,b,c,d)= (a,a,2a,8m),\ (a,3a,8k+2,8m+6),\ (a,3…

Ramanujan's theta functions and linear combinations of three triangular numbers

Let $\Bbb Z$ be the set of integers. For positive integers $a,b,c$ and $n$ let $N(a,b,c;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2$, and let $t(a,b,c;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2$ $(x,y,z\in\Bbb Z)$. In this paper, by using Ramanujan's theta functions $φ(q)$ and $ψ(q)$ we reveal the relation between $t(2,3,3;n)$ and $N(1,3,3;n+1)$, and the relation between $t(1,1,6;n)$ and $N(1,1,3;n+1)$. We also obtain formulas for $t(a,3a,4b;n),$ $t(a,7a,4b;n),t(3a,5a,4b;n)$ and $t(a,15a,4b;n)$ under certain congruence conditions, where $a$…

关于嵌入维数为四的数值半群

On numerical semigroups with embedding dimension four

本文开发了一种几何方法，用于计算任意嵌入维数为四的数值半群的Apéry集，并应用于四个连续平方数和四个连续三角数生成的数值半群，得到了Frobenius数、亏格、Betti元素、极小表示和链环度。