Topic
Squared triangular number
About: Squared triangular number is a research topic. Over the lifetime, 22 publications have been published within this topic receiving 233 citations.
Papers
TL;DR: In this article, it was shown that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is the sum of a triangular number plus a square.
Abstract: By means of $q$-series, we prove that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is a sum of a triangular number plus $x^2+y^2$ for some integers $x$ and $y$ with $x
ot\equiv y (mod 2)$ or $x=y>0$. The paper also contains some other results and open conjectures on mixed sums of squares and triangular numbers.
48 citations
TL;DR: In this article, a generalization of the result rk(8n + k) = cktk(n), which holds for 1 ≤ k ≤ 7, where ck is a constant that depends only on k, is presented.
Abstract: Let rk(n) and tk(n) denote the number of representations of n as a sum of k squares, and as a sum of k triangular numbers, respectively. We give a generalization of the result rk(8n + k) = cktk(n), which holds for 1 ≤ k ≤ 7, where ck is a constant that depends only on k. Two proofs are provided. One involves generating functions and the other is combinatorial.
39 citations
1 Jan 2012
TL;DR: A number N is a square if it can be written as N = n 2 for some natural number n; a triangular number is a triangle if it is written as n(n + 1)/2 for a given natural number N; and a balancing number if 8N 2 + 1 is a squared triangular number.
Abstract: A number N is a square if it can be written as N = n 2 for some natural number n; it is a triangular number if it can be written as N = n(n + 1)/2 for some natural number n; and it is a balancing number if 8N 2 + 1 is a square. In this paper, we study some properties of balancing numbers and square triangular numbers.
35 citations
1 Jan 2006
TL;DR: In this article, the theory of theta functions is used to discover formulas for the number of representations of N as a sum of three squares and for the numbers of representations as sum of triangular numbers.
Abstract: In this note we use the theory of theta functions to discover formulas for the number of representations of N as a sum of three squares and for the number of representations of N as a sum of three triangular numbers. We discover various new relations between these functions and short, motivated proofs of well known formulas of related combinatorial and number-theoretic interest.
15 citations
15 Aug 2006
11 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2016 | 2 |
| 2014 | 1 |
| 2012 | 1 |
| 2009 | 1 |
| 2008 | 4 |
| 2007 | 3 |