Magic Squares A La Fibonacci

For years people have been fascinated with magic squares as well as the Fibonacci sequence of numbers. Put the two together, and you have an especially intriguing mathematical pastime.

In a "regular" magic square, the sum of each row, column, and diagonal is the same number. Consider a regular, 3 x 3 magic square that uses the counting numbers 1 through 9:

8 1 6

3 5 7

4 9 2

The Fibonacci sequence is 1, 2, 3, 5, 8, 13, 21, ... where each term is the sum of the preceding two terms. Take any sequence of nine Fibonaccis and pair them up with the magic square counting numbers 1 through 9. The square below uses the Fibonacci terms 2, 3, 5, 8, 13, 21, 34, 55, 89:

55 2 21

5 13 34

8 89 3

A new magic square is formed, but not with the usual properties. In a Fibonacci magic square, the sum of the products of each row is equal to the sum of the products of each column.
Looking at the rows in the square above,
55 x 2 x 21 = 2310
5 x 13 x 34 = 2210
8 x 89 x 3 = 2136
So, the sum of these products is 2310 + 2210 + 2136 = 6656.
Looking at the columns in the square above,
55 x 5 x 8 = 2200
2 x 13 x 89 = 2314
21 x 34 x 3 = 2142
So, the sum of these products is 2200 + 2314 + 2142 = 6656.

Try the same procedure with a different sequence of nine Fibonaccis. In fact, why not try it with your own sequence, created a la Fibonacci (example: 3, 3, 6, 9, 15, 24, ...)?

To prove this interesting property (and to show that regular magic squares also have the property):

(1) Let x = first Fibonacci number and y = second. Then the Fibonacci sequence would be:
x, y, x + y, x + 2y, 2x + 3y, 3x + 5y, 5x + 8y, 8x + 13y, 13x + 21y.
The sum of the product of the rows = the sum of the product of the columns =
34x³ + 133x²y + 167xy² + 66y³

Fibonacci:

8x + 13y x 3x + 5y

x + y 2x + 3y 5x + 8y

x + 2y 13x + 21y y

(2) Now for the regular magic square,
let x = first number, then x + n = second number, so the sequence looks like this:
x, x + n, x + 2n, x + 3n, x + 4n, x + 5n, x + 6n, x + 7n, x + 8n.

Regular:
The sum of the product of the rows = the sum of the product of the columns =
3x³ + 36x²n + 114xn² + 72n³

x + 7n x x + 5n

x + 2n x + 4n x + 6n

x + 3n x + 8n x + n

(3) Now look at a "multiplicative" magic square:
let a = first number, then ax = second number, ax² = third number, so the sequence looks like this:
a, ax, ax², ax³, ax⁴, ax⁵, ax⁶, ax⁷, ax⁸.

Multiplicative:
The sum of the product of the rows = the sum of the product of the columns = 3a³x¹².

ax⁷ a ax⁵

ax² ax⁴ ax⁶

ax³ ax⁸ ax