Hans Walser, [20150820]

Fibonacci triangle

1     The triangle

Figure 1 gives the Fibonacci triangle.


Fig. 1: Fibonacci triangle


In Figure 2 we see an orthogonal arrangement.


Fig. 2: Orthogonal arrangement


2     Notation

For the Fibonacci numbers we use the usual upper case notation:




For the Fibonacci triangle we will use the lower case notation of Figure 3.


Fig. 3: Notation


3     Features of the Fibonacci triangle

3.1    The Fibonacci numbers

The numbers in the first column of Figure 2 are the usual Fibonacci numbers:




The Fibonacci numbers appear also under the roof:




3.2    Signs

The signs change every second step (Fig. 4).


Fig. 4: Signs


3.3    Symmetry

The Fibonacci triangle is not symmetric. But leaving away the ŇroofÓ on the right and ignoring the signs there is an axial symmetry (Fig. 5).


Fig. 5: Quasi symmetry


4     Powers of the Fibonacci numbers

4.1    Squares and cubes

Lets begin with a common picture (Fig. 6). It depicts the growth of the Fibonacci numbers. Beginning with a unit square we get the Fibonacci numbers for the side lengths of the square sequence. And of course we have the recursion:




Fig. 6: Fibonacci squares

For the areas of the squares we obtain the sequence :


                                            1, 1, 4, 9, 25, 64, 169, 441, 1156, ...                                       (5)


Playing with these numbers we find a three digits recursion:








That means, that in Figure 6 the green and the cyan square have together half the area of the yellow plus the blue square. This can be seen by the dissection proof of Figure 7.

Fig. 7: Dissection


Notice that the coefficients of (6) appear in the Fibonacci triangle:




For the cubes  of the Fibonacci numbers we have:


                                      1, 2, 8, 27, 125, 512, 2197, 9261, 39304, ...                                 (9)


Here we find a four digits recursion:








4.2    General powers

For the kth power of the Fibonacci numbers there is a (k + 1) digits recursion:




For the proof of (12) I used a computer algebra system.

5     The elements of the Fibonacci triangle

5.1    Columns

We have already found (2) the Fibonacci numbers in the first column of Figure 2.

In the next column of Figure 2 we have the products of two consecutive Fibonacci numbers:








Again in the next column we get products of three consecutive Fibonacci numbers, but with a negative sign an a coefficient of one half:








5.2    Overview

For the first five columns we get:



The denominators in the coefficients are the products of consecutive Fibonacci numbers beginning with  (kind of ŇfactorialÓ):  




5.3    General

Finally we get the general formula for the elements of the Fibonacci triangle:




6     Recursion in the Columns

The elements in the columns of the Fibonacci triangle fulfill a recursion like the recursion (12) for the powers of the Fibonacci numbers. Id est:





First example: n = 7, k = 1




Figure 8 shows the involved elements. Red is the dot product of cyan and yellow.


Fig. 8: Dot product


Second example: n = 8, k = 2




In Figure 9 the involved elements:


Fig. 9: Involved elements