Tetrahedral 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.

Abstract: Notation: Z the set of integers, N the set of positive integers, [x] the greatest integer not exceeding x, ( a m) the Legendre-JacobiKronecker symbol, ordpm the nonnegative integer α such that pα | m but pα+1 m, (a, b) the greatest common divisor of a and b, (a, b, c) the form ax2 + bxy + cy2, [a, b, c] the equivalence class containing the form (a, b, c), H(d) the form class group of discriminant d, h(d) the class number of discriminant d, R(K, n) the number of representations of n by the class K.

Abstract: A two-dimensional Euclidean plane can be identified with the Gauss-Argan plane, so that a two-dimensional triangular lattice can be identified, apart from a scaling factor, with a set of 'triangular integers' of the complex quadratic field associated with a complex cubic root of 1. It is shown that a two-dimensional modulated structure on the lattice can be characterised by a 'rational triangular number' of the quadratic field. Possible modulated structures are shown to be determined by the number-theoretical properties of the 'triangular integers'. The continued-fraction-expansion algorithm for ordinary rational numbers is extended to the case of 'rational triangular numbers'. Commensurate structures on the triangular lattice are classified by the orders of the continued-fraction expansions of the relevant 'rational triangular numbers'. The atomic configuration in a commensurate structure of a system of atoms adsorbed on a substrate with a triangular lattice is shown to have a hierarchical structure corresponding to the hierarchy among the convergents to the relevant 'rational triangular numbers'.

Abstract: A celebrated result of Gauss states that every positive integer can be represented as the sum of three triangular numbers. In this article we study p3∆(n), the number of partitions of the integer n into three triangular numbers, as well as p3∆(n), the number of partitions of n into three distinct triangular numbers. Unlike t(n), which counts the number of representations of n into three triangular numbers, p3∆(n) and p d 3∆(n) appear to satisfy very few arithmetic relations (apart from certain parity results). However, we shall show that, for all n ≥ 0, p3∆(27n+ 12) = 3p3∆(3n+ 1) and p d 3∆(27n+ 12) = 3p d 3∆(3n+ 1). Two separate proofs of these results are given, one via generating function manipulations and the other by a combinatorial argument.