Fibonacci
Calculation methods (Part 3)
We are looking for expressions for sums of powers of Fibonacci numbers, thus for
g1,n = F1 + F2 + F3 + ... + Fn
g2,n = F12 + F22 + F32 + ... + Fn2
g3,n = F13 + F23 + F33 + ... + Fn3
etc.
We start with g1,n:
+g1,n = F1 + F2 + F3 + F4 + F5 + ... + Fn-2 + Fn-1 + Fn
-g1,n = F1 - F1 - F2 - F3 - F4 - F5 - ... + - Fn-2 - Fn-1 - Fn
-g1,n = F1 - F2 - F1 - F2 - F3 - F4 - F5 - ... + + - Fn-2 - Fn-1 - Fn
Count together the terms below every column and remember that
Fk+1 - Fk - Fk-1 = 0 for every k.
This yields:
-g1,n = F1 + (F2 - F1) + 0 + ... + 0 + (- Fn - Fn-1) - Fn.
or: -g1,n = 1 -Fn+1 - Fn = 1 - Fn+2.
So g1,n = Fn+2 - 1.
For g2,n we try to find a similar approach.
Thereto we need a relation between squares of Fibonacci numbers. Such a relation does exist:
Fp+32 - 2Fp+22 - 2Fp+12 + Fp2 = 0. (see Some general formulas)
This leads us to:
.+g2,n = F12 + F22 + F32 + F42 + ... + Fn2
-2g2,n = ++++-2F12 -2F22 -2F32 -2F42 - ... -2Fn2
-2g2,n = ++++++++++-2F12 -2F22 -2F32 -2F42 - ... -2Fn2
.+g2,n =++++++++++++++++++ F12 + F22 + F32 + F42 + ... + Fn2
Again, add the numbers in every column and use the just mentioned relation. This yields:
-2g2,n = F12 + (F22 - 2F12) + (F32 - 2F22 - 2F12) + 0 + ... + 0 +
(-2Fn2 - 2Fn-12 + Fn-22) +
(-2Fn2 + Fn-12) + Fn2,
or: -2g2,n = -3F12 - F22 + F32 + Fn-22 - Fn-12 - 3Fn2 = Fn-22 - Fn-12 - 3Fn2
The expression can slightly be simplified.
Fn-22 - Fn-12 - 3Fn2 =
(Fn - Fn-1)2 - Fn-12 - 3Fn2 =
Fn2 + Fn-12 -2FnFn-1 - Fn-12 - 3Fn2 =
-2FnFn-1 - 2Fn2 =
-2Fn(Fn-1 + Fn) = -2FnFn+1.
Thus g2,n = FnFn+1.
For g3,n we find in chapter "Some general formulas" (with p=q=r) the following relation between cubes:
Fp+43 - 3Fp+33 - 6Fp+23 + 3Fp+13 + Fp3 = 0
The beforementioned method leads to the next result:
-4g3,n = F13 + (F23 - 3F13) +
(F33 - 3F23 - 6F13) +
(F43 - 3F33 - 6F23 + 3F13) + 0 + ... + 0 +
(-3Fn3 - 6Fn-13 + 3Fn-23 + Fn-33) +
(-6Fn3 + 3Fn-13 + Fn-23)
+ (3Fn3 + Fn-13) + Fn3,
ofwel: -4g3,n = -2 - 5Fn3 - 2Fn-13 + 4Fn-23 + Fn-33
The hobbyist may try to proof that this formula can also be written as (use the formula for Fn from A more general formula for Fn):
10g3,n = 5 + F3n+2 - 6(-1)nFn-1.