TIGHT UNIVERSAL TRIANGULAR FORMS

For a subset S of nonnegative integers and a vector $\mathbf {a}=(a_1,\ldots ,a_k)$ of positive integers, define the set $V^{\prime }_S(\mathbf {a})=\{ a_1s_1+\cdots +a_ks_k : s_i\in S\}-\{0\}$ . For a positive integer n, let $\mathcal T(n)$ be the set of integers greater than or equal to n. We consider the problem of finding all vectors $\mathbf {a}$ satisfying $V^{\prime }_S(\mathbf {a})=\mathcal T(n)$ when S is the set of (generalised) m-gonal numbers and n is a positive integer. In particular, we completely resolve the case when S is the set of triangular numbers.

Triangle of Triangular Numbers

Among several interesting number triangles that exist in mathematics, Pascal’s triangle is one of the best triangle possessing rich mathematical properties. In this paper, I will introduce a number triangle containing triangular numbers arranged in particular fashion. Using this number triangle, I had proved five interesting theorems which help us to generate Pythagorean triples as well as establish bijection between whole numbers and set of all integers.

Fuzzy-Based EOQ Model With Credit Financing and Backorders Under Human Learning

In this paper, an inventory model has been developed with trade credit financing and back orders under human learning. In this model, it is considered that the seller provides a credit period to his buyer to settle the account and the buyer accepts the credit period policy with certain terms and conditions. The impact of learning and credit financing on the size of the lot and the corresponding cost has been presented. For the development of the model, demand and lead times have been taken as the fuzzy triangular numbers are fuzzified, and then learning has been done in the fuzzy numbers. First of all, the consideration of constant fuzziness is relaxed, and then the concept of learning in fuzzy under credit financing is joined with the representation, assuming that the degree of fuzziness reduces over the planning horizon. Finally, the expected total fuzzy cost function is minimized with respect to order quantity and number of shipments under credit financing and learning effect. Lastly, sensitive analysis has been presented as a consequence of some numerical examples.

Congruence properties of indices of triangular numbers multiple of other triangular numbers

For any non-square integer multiplier \(k\), there is an infinity of triangular numbers multiple of other triangular numbers. We analyze the congruence properties of indices \(\xi\) of triangular numbers multiple of triangular numbers. Remainders in congruence relations \(\xi\) modulo \(k\) come always in pairs whose sum always equal \((k-1)\), always include 0 and \((k-1)\), and only 0 and \((k-1)\) if \(k\) is prime, or an odd power of a prime, or an even square plus one or an odd square minus one or minus two. If the multiplier \(k\) is twice the triangular number of \(n\), the set of remainders includes also \(n\) and \((n^{2}-1)\) and if \(k\) has integer factors, the set of remainders include multiples of a factor following certain rules. Algebraic expressions are found for remainders in function of \(k\) and its factors, with several exceptions. This approach eliminates those \(\xi\) values not providing solutions.

Binomial Coefficients and Triangular Numbers

We produce formulas of sums the product of the binomial coefficients and triangular numbers. And we apply our formula to prove an identity of Wang and Zhang. Further, we provide an analogue of our identity for the alternating sums.

A sum of three nonzero triangular numbers

Finding all integers which can be written as a sum of three nonzero squares of integers has been studied by a number of authors. This question is solved under the assumption of the Generalized Riemann Hypothesis (GRH), but still remains unsolved unconditionally. In this paper, we show that out of all integers that are sums of three squares, all but finitely many can be written as [Formula: see text] for some integers [Formula: see text]. Furthermore, we explicitly describe this finite set under the GRH. From this result, we also describe further generalizations for sums of nonzero polygonal numbers. Precisely, we find all integers, under the GRH only when [Formula: see text], which are sums of [Formula: see text] nonzero triangular (generalized pentagonal and generalized octagonal, respectively) numbers for any integer [Formula: see text].