Finding a pattern in a sequence of numbers and describing that pattern recursively and with an explicit formula are common mathematical activities. Rather than starting with a sequence and then analysing it to discover patterns, we started with particular patterns then constructed sequences that were built from those patterns. As we engaged in geometric and algebraic explorations of that process, we were fascinated by ways in which we could reveal the triangular numbers as 'steps' in the construction of quadratic and cubic sequences. In this article, we share highlights of those explorations, which include using triangular numbers to derive formulas for quadratic and cubic sequences and constructing a 2-dimensional geometric proof of the formula for the nth tetrahedral number.