Juxtapositions: Fibonacci’s Triangle

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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.

Pascal’s Triangle 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.

Fibonacci’s Triangle

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.

Sierpinski’s TriangleThere 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!

*You can see the picture in a 1665 copy of Pascal’s book Traité du triangle arithmétique here; other images are available from the Cambridge University Library Digital Image Collection.

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4 Responses to “Juxtapositions: Fibonacci’s Triangle”

  1. TwoPi Says:

    Way cool.

    In the “Pascal” triangle, each row sums to a power of two.

    In the Fibonacci triangle, the row sums are 1, 2, 6, 14, 32, 70, 150, 316, …
    I don’t see an obvious pattern there, so I punted and handed it off to the On-line Encyclopedia of Integer Sequences. I got all excited when I got a hit, but lo and behold the matching sequence A074878 is described as being the row sums of the Fibonacci Triangle.

    I guess that’s a sign that there are theorems lurking here to discover and publish.

  2. Juxtaposition: Millionaire Triangles « 360 Says:

    […] got me thinking, what would the other triangles look like?  We’ve actually posted before on Fibonacci’s Triangle:  as envisioned by Doug Ensley, it’s a triangle with Fibonacci numbers down each side and […]

  3. rajesh Says:

    can i will be able to use this sequence in my day to day life for trading in shares or use this sequence to gain extra income, if yes how i am being thinking a lot since two months now.

  4. Ξ Says:

    Rajesh, certainly you can use this sequence for trading, but it’s no more likely to get you extra income than it is to cause you to lose money. [Hmm. That’s assuming a zero-sum game. These days, no matter what you do you might be somewhat more likely to lose money than to gain it, at least in the short term.]

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