Investigating Triangular Numbers with greatest integer function, Sequences and Double Factorial

The Triangular number denoted by is defined as the sum of the first consecutive positive integers. A positive integer is a Triangular Number if and only if [1]. We stated and proved a sequence of positive integers is consecutive triangular numbers if and only if √ − √ =1 and √ . We consider a ceiling function ⌈ ⌉ to state and prove a necessary and sufficient condition for a number ⌈ ⌉ ⌈ ⌉ to be a triangular number for each . A formula to find and of any two consecutive triangular numbers and a double factorial is introduced to find products of triangular numbers.
Key words: Triangular numbers, ceiling function, double factorial.