• School Mathematics
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• v.12 no.4
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• pp.619-638
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• 2010

In this paper, we listed and reviewed some properties on polygonal numbers, pyramidal numbers and Pascal's triangle, and Fibonacci sequence. We discussed that the properties of gnomonic numbers, polygonal numbers and pyramidal numbers are explained integratively by introducing the generalized k-dimensional pyramidal numbers. And we also discussed that the properties of those numbers and relationships among generalized k-dimensional pyramidal numbers, Pascal's triangle and Fibonacci sequence are suitable for teaching and learning of mathematical reasoning and connections.

• Journal of the Korean Mathematical Society
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• v.48 no.4
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• pp.837-847
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• 2011

For an integer $m{\geq}3$, every integer of the form $p_m(x)$ = $\frac{(m-2)x^2(m-4)x}{2}$ with x ${\in}$ $\mathbb{Z}$ is said to be a generalized m-gonal number. Let $a{\leq}b{\leq}c$ and k be positive integers. The quadruple (k, a, b, c) is said to be universal if for every nonnegative integer n there exist integers x, y, z such that n = $ap_k(x)+bp_k(y)+cp_k(z)$. Sun proved in [16] that, when k = 5 or $k{\geq}7$, there are only 20 candidates for universal quadruples, which h listed explicitly and which all involve only the case of pentagonal numbers (k = 5). He veri ed that six of the candidates are in fact universal and conjectured that the remaining ones are as well. In a subsequent paper [3], Ge and Sun established universality for all but seven of the remaining candidates, leaving only (5, 1, 1, t) for t = 6, 8, 9, 10, (5, 1, 2, 8) and (5, 1, 3, s) for s = 7, 8 as candidates. In this article, we prove that the remaining seven quadruples given above are, in fact, universal.