The Pythagonacci Connection

I've added a couple of links at the bottom to Professor Kalman's original presentation visuals and an article he co-wrote that explains much more about the idea, some of the history, and some earlier citations on it that I want to pursue.  As I get through some of these, I will try to link those that I think would be helpful to HS teachers, or of general interest.  


Just so you know, I'm not clever enough to come up with a compound word like Pythagonacci, but I hang out with some clever guys who are, and this one was the creation of Dan Kalman, who has shared his knowledge and indulged my questions for a long time.  Looking over his web page after a recent communication, I found his notes from a presentation on the topic and, I admit it, I'm a sucker for a really cool historical blend.
As often happens with Dan's presentations, you don't have to be very clever to figure out that something VERY clever is happening in front of you.  In this case it was a transformation that took the Fibonacci numbers, and transformed them into the side lengths of a Pythagorean Triangle.  Yeah, that got your attention, huh.
So how does it work, and like me, you may ask yourself, "How come I never noticed that?".

So the basics is you start with any four Fibonacci numbers, I'll use 3, 5, 8., 13 as my selection (that's cause he did, and the arithmetic is there for me to check.).

Now he calls his method OTIFAL, which I take to stand for Outsides, Twice Insides, and First and Last So let's look at the four numbers as if they were a binomial, (3,  5) ( 8, 13)  and if we go with outsides, and multiply, we get 3*13 = 39.... write that down, that's one side of our Pythagorean, or Pythagonacci triangle.
Now we do inside both ways, 5*8+8*5 = 80, that's another leg.
Finally we find the hypotenuse with the sum of the product of the firsts and the lasts, 3*8 + 5*13 = 89.
And sure enough, We have a Pythagonacci right triangle (39, 80, 89)  .

A couple of quick notes from my first reading of Professor Kalman's work with Professor Mena, is that he points out that all those sweet mysteries of the Fibonacci and Lucas sequences are not simply some Joint Wonderkind, but part of a general pattern of sequences.  Here, in their own words, " These are sequences An defined by a recursive rule An+2 = a An+1 + b An where a and b are fixed constants. We refer to such a sequence as a two-term recurrence".  He then points out ten characteristic aspects of the Fibonacci-Lucas sequences that are preserved by these two term recurrences.  These include the fact that :

The sum of the squares of the first n Fibonacci numbers, are given by the product of \(F_n * F_{n+1}\) 

So, for example, \(1^2 + 1^2+ 2^2 + 3^2 + 5^2 = 5 x 8\)

The fact that the sequence contains its own running sum, and ending with the facts that they also preserve the Pythagorean Theorem property (he actually gives a shorter one), and the fact that the GCD of two numbers in the sequence has the index of the GCD of the indices of the two sequence values.


Here are just some of the patterns that emerge, including: many you may recognize, but (like me) never thought of them as Fibonacci-like.
If you start with a Rule like 3 S(n-1) - 2 (Sn-2) on a sequence 0, 1, you get 3, 7, 15... and quickly relaize you are creating the Mersenne sequence \( 2^n-1\) .  If you start with the Lucas beginning with 2, 3 the same rule continues 5, 9, 17... and you are getting the Fermat Sequence, \( 2^n+1\) . They give another that produces the Pell-Lucas numbers, and one that gives only the even Fibonacci and Lucas numbers with even index.

And the Pythagonacci connection, they show that you don't need four numbers, since you can start with two, and create the outside two.  So in our sequence above with 3, 5, 8, 13, the three is their difference, and the 13 is their sum.  We could have just used \(8^2 - 5^2\), 2*5*8 stays the same,  and \(5^2 + 8^2\) to produce the 39, 80, 89 from before.  From any two consecutive Fibonacci numbers, we may fall back to the method often credited to Euclid, but certainly known well before 300 BC,  \(x^ 2 − y^2\), 2xy, \( x^ 2 + y^2\). is the whole ticket.  Still, if I were teaching this to young people, I would probably give them the four number approach, and wait.... maybe one of them is going to look, converse with another, and a hand goes up.... But Mr. Ballew, couldn't you ...And I will never deny them that.

I couldn't get access to the oldest article cited in the Professors' paper, but I found another on his list by William Boulger that gives lots of information about the earlier paper, and credits the earliest observation of this Pythagonacci relationship to Charles W. Raine, and a paper in Scripta Mathematica in 1948. . Charles W. Raine, Pythagorean triangles from the Fibonacci series, Scripta Mathematica 14 (1948), 164, I could not find an active link to this. 
I will also shocked to read that Boulger noted that the fact appeared in the 1986 Penguin Dictionsary of Curious and Interesting Numbers, by David Wells.  Shocked because it was within four feet of my desk, and I had read through it dozens of times and NEVER SAW THAT!!!!.  I just checked, it is there, and credits Raine with the observation as well. It also points out that the Area of the right triangle formed is the product of the four Fibonacci numbers.  In the case of 3, 5, 8, 13 the area is 1560sq units. It also points out, as did Boulger, that the hypotenuse has an index that is half the sum of the indices of the four original numbers, or just the sum of the middle two.  In my example the numbers are the 4th, 5th, 6th, and 7th Fibonacci numbers, and 89 is the 11th (5+6) or 1/2(4+5+6+7).  . 

Boulger's article is available on JSTOR at William Boulger, Pythagoras Meets Fibonacci, Mathematics Teacher 82 (1989), 277–782. or perhaps at your local college library. 

The papers below are incredible, and every MS or HS teacher will surely find morsels for generating additional classroom interest.  I'm going to work my way through as many of the references in there to see if I can find the first observation of the Pythagonacci connection.  That ought to be more common knowledge.  Thanks Again to Professor Kalman for sharing this.   .


Professor Kalman pointed out where I had overlooked the article which provided more context on the problem, so here are the links.  The Joint paper (about 17 pages) with Robert Mena , and the original slides I found to write the first part of this post.