Fibonacci

Some general formulas (part 2)

Sum and difference of products.

On the previous page we saw formula 3.):
F_p+iF_q+i - (-1)ⁱF_pF_q = F_iF_p+q+i
Let us write this formula more symmetrical.
This can be done by replacing i by 2i and further p and q by (respectively) p-i and q-i. This yields:
F_p+iF_q+i - F_p-iF_q-i = F_2iF_p+q
Let us try to generalize these two formulas and find a formula in which the terms
F_p+iF_q+iF_r+i,      F_p-iF_q-iF_r-i and F_p+q+r are present. This yields the following variant:

19.) F_p+iF_q+iF_r+i - (-1)ⁱL_iF_pF_qF_r + (-1)ⁱF_p-iF_q-iF_r-i = F_iF_2iF_p+q+r (proof)
For i=1 we have a nice formula
20.) F_p+1F_q+1F_r+1 + F_pF_qF_r - F_p-1F_q-1F_r-1 = F_p+q+r
The corresponding Lucas variant is:
21.) L_p+iL_q+iL_r+i + 25(-1)ⁱF_iF_pF_qF_r - (-1)ⁱL_p-iL_q-iL_r-i = 5L_iF_2iF_p+q+r
and the corresponding mixed variants are:
22.) L_p+iF_q+iL_r+i + 5(-1)ⁱF_iF_pL_qF_r - (-1)ⁱL_p-iF_q-iL_r-i = L_iF_2iL_p+q+r
23.) F_p+iL_q+iF_r+i - (-1)ⁱL_iF_pL_qF_r + (-1)ⁱF_p-iL_q-iF_r-i = F_iF_2iL_p+q+r

Some other generalizations.

The formula F_p+2 = F_p+1 + F_p can be generalized as follows:
24.) F_p+3F_q+3 = 2F_p+2F_q+2 + 2F_p+1F_q+1 - F_pF_q and
25.) F_p+4F_q+4F_r+4 = 3F_p+3F_q+3F_r+3 + 6F_p+2F_q+2F_r+2 - 3F_p+1F_q+1F_r+1 - F_pF_qF_r.
See also next page.

F₁ + F₂ + F₃ + ... + F_n = F_n+2 - 1 and
F₁ + F₃ + F₅ + ... + F_2*n+1 = F₁ + (F₁ + F₂) + (F₃ + F₄) + ... + (F_2*n-1 + F_2*n) = 1 + F_2*n+2 - 1 = F_2*n+2
At large {{ You may proof this by induction with variable k and using. formula 1 }}:
26.) F_a + F_a+r + F_a+2r + ... + F_a+kr = ((-1)^rF_a+kr - F_a+(k+1)r + (-1)^aF_r-a + F_a)/(1 - L_r + (-1)^r).