Triangular Numbers

This study established some recurrence relations and exponential generating functions of the sequence of factoriangular numbers. A factoriangular number is defined as a sum of corresponding factorial and triangular number. The proofs utilize algebraic manipulations with some known results from calculus, particularly on power series and Maclaurin's series. The recurrence relations were found by manipulating the formula defining a factoringular number while the ascertained exponential generating functions were in the closed form.

A new sequence of natural numbers can be formed by adding corresponding factorials and triangular numbers. In this paper, such numbers were named factoriangular numbers. Mathematical experimentations on these numbers resulted to the establishment of some of its characteristics. These include the parity, compositeness, the number and sum of its positive divisors, abundancy and deficiency, Zeckendorf's decomposition, end digits, and digital roots of factoriangular numbers. Several theorems and corollaries were proven and some conjectures were also presented.

The study of heat transfer characterstics in micro channel heat sinks is an emerging field from the research point of view. The variations in shapes and sizes of micro channel heat sink has led to the increase in heat transfer rate and reduction in pressure drop. Earlier work has been performed on different cross-sections of micro channel heat sink using water as a fluid for heat transfer. The numerical computation is carried out by solving the governing equations along with boundary conditions .The equations for fluid in single phase was solved by using a single domain conjugate problem. The governing equations is discretize in the fluid region using finite-volume method (FVM) with hybrid differencing scheme. The flow field is solved using the Simplex algorithm. The design of the micro-heat sink consists of an array of 11 holes cut in a channel of length 22mm, width 12mm and height 1.5mm. In this paper, we analytically designed a micro-channel heat sink for specific boundary conditions of inlet velocity of 0.1m/s, fluid temperature of 278K and temperature on the left side of 328 K and right side to be adiabatic. The analysis is performed numerically for its validation. The design was tested with different cross sections like rectangular, triangular, hexagonal ,circular and also compared the fluid for no. of channels 9,18 and 27 by using n-octane for heat transfer coefficient at reynold number range 100-1000.

The nth triangular number, denoted by T (n), is defined as the sum of the first consecutive positive integers. It can be represented in the form of a triangular grid of points. As a result, this study discusses the triangular numbers and its properties. To distinguish whether a positive integer N is a triangular number or not, it has to comply with 8N + 1 which is a perfect square; characteristics of odd and even triangular numbers; sum of two consecutive triangular numbers with same parity i.e., the formula for 2

A new sequence of natural numbers can be formed by adding corresponding factorials and triangular numbers. In this paper, such numbers were named factoriangular numbers. Mathematical experimentations on these numbers resulted to the establishment of some of its characteristics. These include the parity, compositeness, the number and sum of its positive divisors, abundancy and deficiency, Zeckendorf's decomposition, end digits, and digital roots of factoriangular numbers. Several theorems and corollaries were proven and some conjectures were also presented.