ANSWER: MAGIC SQUARES AND FIBONACCI NUMBERS:

Yes, I have some new information about Magic Squares made up from the Fibonacci Numbers:
It comes from a beautiful book that you must surely have:

"The Divine Proportion", subtitled:

A Study in Mathematical Beauty by H. E Huntley

(Dover Publication, 1970, New York, isbn: 486-22254-3)

On page 124 to 126.

Its actually a Multiplication Square using Fibonacci Numbers, but still very very interesting.

Firstly, you know of Multiplication Squares, the simplest one being the Magic Square of 3x3 where the numbers are replaced by the powers of 2:

128  1   32

4    16   64

8    256  2

The Magic Sum, as you know is 4,096 and each cell contains a power of 2 from 2 to the zero = 1,  to 2 to the eighth power = 256, the indices being the numbers found in the numbers:

8  1  6

3  5  7

4  9  2

Now, a sort of Magic Square can be formed with any consecutive members of the Fibonacci series. An example will make this clear.

Take any 9 consecutive numbers from the following series:

           v1  v2  v3  v4  v5   v6   v7  v8  v9

0  1  1  2    3    5    8   13   21   34  55  89  144  233  377...

Now form a Magic Square in which the subscripts of "v" add up to 15:

v8  v1  v6

v3  v5  v7

v4  v9  v2

Substitute for these the following Fibonacci numbers:

89   3   34

8    21  55

13  144  5

The sum of the products of the three rows is 9078 + 9240 + 9369

which = 27,678.

The sum of the products of the three columns is 9256 + 9072 + 9350

which = 27,678

Hope this is of use to you. Its not a true Magic Square, but is quite elegant and has its own Consciousness, it is a new entity and perhaps you can give it life.

Let me know your response to this.

Perhaps it will inspire you to find a true Fibonacci Magic Square or Cube... 

In Europe, Fibonacci’s Liber Abaci in 1202 described these numbers; the book was meant to introduce the  Indian number system and its mathematics which he had learnt in North Africa from Arab teachers.

While a young man growing up there, Fibonacci speaks of his education in North Africa thus: (“):

My father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it. 

It has been suggested that the name Gopala-Hemachandra numbers be used for
the general sequence: a, b, a+b, a+2b, 2a+3b, 3a+5b, ... for any pair a, b, which for the case a=1, b=1 represents the Fibonacci numbers. This would then include the Lucas series for which a=2 and b=1.
It would also include other series such as the one for which a=1 and b=21, which generates
the numbers: 1, 21, 22, 43, 65, 108, (...);. Narayana Pandita (’);s Ganita Kaumudi (1356) studies additive sequences where each term is the sum of the last q terms. He poses the problem thus: (“);
A cow gives birth to a calf every year. The calves become young and they begin giving birth to calves when they are three years old. Tell me, O learned man, the number of progeny produced during twenty years by one cow. (”); (See Singh, 1936; and Singh, 1985 for details).

Taken from: