What happens when you take two different ideas and put them together? That’s a driving question in mathematical research, and it can lead to some interesting and entertaining results.
One such juxtaposition is the concept of Fibonacci’s Triangle: a combination of the Fibonacci Sequence (1, 1, 2, 3, 5, 8, … where each number is the sum of the previous two) and Pascal’s Triangle*, shown to the left, which is created by placing “1”s along the outside and then filling the inside by adding two adjacent numbers and placing the sum between them on the next row.
In Fibonacci’s Triangle, according to Doug Ensley (see below), the string of “1”s along the outside is replaced by the Fibonacci sequence. The inside is filled in the same way as in Pascal’s triangle, with adjacent numbers in a row added together and the sum placed between them in the next row.
There are many questions that can be asked about Fibonacci’s triangle: Is there an explicit formula for the entries in each row the way there is for Pascal’s triangle? Does a Sierpinski-like pattern develop when you shade in the odd numbers the way it does for Pascal’s triangle? The answer to the first question is yes: see Doug Ensley’s article, “Fibonacci’s Triangle and Other Abominations” in the September 2003 issue of Math Horizons. For the second question, the best solution is to draw it out and see for yourself!