Fibonacci

Some general formulas (part 1)

Sum en difference of squares.

F_n+k² - F_n-k² = F_2nF_2k
In general there is no simple expression for the sum F_n+k² + F_n-k².
F_n+k+1² + F_n-k² = F_2n+1F_2k+1
For the difference F_n+k+1² - F_n-k² we know the following expressions for k=0 and k=1
1.) F_n+1² - F_n² = F_n-1F_n+2 and
2.) F_n+2² - F_n-1² = 4F_nF_n+1


Sum and difference of products.

The first two formulas of this page are special cases of the formula:
3.) F_p+iF_q+i - (-1)ⁱF_pF_q = F_iF_p+q+i
The corresponding Lucas variant is:
4.) L_p+iL_q+i - (-1)ⁱL_pL_q = 5F_iF_p+q+i
and the corresponding mixed variants are:
5.) F_p+iL_q+i - (-1)ⁱF_pL_q = F_iL_p+q+i and
6.) F_p+iL_q+i + (-1)ⁱL_pF_q = L_iF_p+q+i and
7.) 5F_p+iF_q+i + (-1)ⁱL_pL_q = L_iL_p+q+i and
8.) L_p+iL_q+i + 5(-1)ⁱF_pF_q = L_iL_p+q+i


Another similar variant is
9.) F_p+iF_q - F_pF_q+i = (-1)^pF_iF_q-p
It Lucas counterpart is:
10.) L_p+iL_q - L_pL_q+i = -5(-1)^pF_iF_q-p
The mixed forms are:
11.) F_p+iL_q - F_pL_q+i = (-1)^pF_iL_q-p and
12.) F_p+iL_q - L_pF_q+i = -(-1)^pL_iF_q-p and
13.) 5F_p+iF_q - L_pL_q+i = L_iL_q-p and
14.) L_p+iL_q - 5F_pF_q+i = -L_iL_q-p

More general

A.) F_p+iF_q-i - F_pF_q = (-1)^k(F_p+i+kF_q-i+k - F_p+kF_q+k) (proof)
B.) L_p+iL_q-i - L_pL_q = (-1)^k(L_p+i+kL_q-i+k - L_p+kL_q+k)
C.) F_p+iL_q-i - F_pL_q = (-1)^k(F_p+i+kL_q-i+k - F_p+kL_q+k)
D.) L_p+iL_q-i - 5F_pF_q = (-1)^k(L_p+i+kL_q-i+k - 5F_p+kF_q+k)
E.) F_p+iL_q-i - L_pF_q = (-1)^k(F_p+kL_q+k - L_p+i+kF_q-i+k)

Sum, difference and double formulas.

Formula 6.) with i=0 yields a wellknown sum formula for F:
15.) F_pL_q + L_pF_q = 2F_p+q
Formula 12.) with i=0, or formula 2:
15a.) F_p L_q - L_pF_q = 2 (-1)^qF_p-q
Formula 8.) with i=0 yields a wellknown sum formula for L:
16.) 5F_pF_q + L_pL_q = 2L_p+q
Formula 13.) with i=0:
16a.) 5F_pF_q - L_pL_q = 2L_q-p
Formula 15.) with q=p yields the famous double formula for F:
17.) F_2p = F_pL_p
Formula 7.) with i=0 and p=q=m yields: 5F_m² + L_m² = 2L_2m and
formula 4.) with i=m and p=q=0 yields: L_m² - 4(-1)^m = 5F_m²
Eliminate 5F_m² from these two equations. The result is the wellknown double formula for L:
18.) L_2m = L_m² - 2(-1)^m