Hans Walser, 

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: (1)

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: (2)

The Fibonacci numbers appear also under the roof: (3)

## 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 “roof” 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: (4) 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: (6)

Example: (7)

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: (8)

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: (10)

Example: (11)

## 4.2    General powers

For the kth power of the Fibonacci numbers there is a (k + 1) digits recursion: (12)

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: (13)

Example: (14)

Again in the next column we get products of three consecutive Fibonacci numbers, but with a negative sign an a coefficient of one half: (15)

Example: (16)

## 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 “factorial”): (17)

## 5.3    General

Finally we get the general formula for the elements of the Fibonacci triangle: (18)

# 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: (19)

Examples:

First example: n = 7, k = 1 (20)

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