1
2; This Mathomatic input file contains the mathematical formula to
3; directly calculate the "n"th Fibonacci number.
4; The formula presented here is called Binet's formula, found at
5; http://en.wikipedia.org/wiki/Fibonacci_number
6;
7; The Fibonacci sequence is the endless integer sequence:
8; 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ...
9; Any Fibonacci number is always the sum of the previous two Fibonacci numbers.
10;
11; Easy to understand info on the golden ratio can be found here:
12; http://www.mathsisfun.com/numbers/golden-ratio.html
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14-1/phi=1-phi ; Derive the golden ratio (phi) from this quadratic polynomial.
150 ; show it is quadratic
16unfactor
17solve verifiable for phi ; The golden ratio will help us directly compute Fibonacci numbers.
18replace sign with -1 ; the golden ratio constant:
19fibonacci = ((phi^n) - ((1 - phi)^n))/(phi - (1 - phi)) ; Binet's Fibonacci formula.
20eliminate phi ; Completed direct Fibonacci formula:
21simplify ; Note that Mathomatic rationalizes the denominator here.
22for n 1 20 ; Display the first 20 Fibonacci numbers by plugging in values 1-20:
23; Note that this formula should work for any positive integer value of n.
24