Fibonacci
Some general formulas (part 3)
Special cases.
When we substitute r=q=p and i=1 in 19.) and 24.), then this leads up to interesting formulas:
F3p = Fp+13 + Fp3 - Fp-13 and
Fp+32 - 2Fp+22 - 2Fp+12 + Fp2 = 0.
Formula 1 of "A special property" can be rewritten (with odd q) as
Fp+2q - Fp = LqFp+q
Now Fp+2q - Fp = Fp-1 + Fp + ... + Fp+2q-2, so (q=3, p=n+1):
The sum of 6 consecutive Fibonacci numbers Fn + Fn+1 + ... + Fn+5 equals 4Fn+4
and (q=5, p=n+1) Fn + Fn+1 + ... + Fn+9 = 11Fn+6
Varia.
(Lm/2 + 
5 Fm/2)(Ln/2 + 5 Fn/2) =
Lm+n/2 + 5 Fm+n/2 and
L0/2 + 5 F0/2 = 1, thus:
The set {Lm/2 + 5 Fm/2) | m integer} constitutes (with the usual multiplication) a multiplicative group.
Define F!m = F1F2...Fm for m>0 and F!0=1 and
Fn,m = F!n/(F!mF!n-m),
then Fn,m is an integer for n≥m≥0, and
2Fn,m = LmFn-1,m + Ln-mFn-1,m-1
A generalization of the formulas 24.) and 25.) of the previous page.
Fp1+mFp2+m...Fpm-1+m = Fm,m-1Fp1+m-1Fp2+m-1...Fpm-1+m-1 + ...
+(-1)[(m-k-1)/2] Fm,kFp1+kFp2+k...Fpm-1+k +...
(-1)[(m-1)/2] Fm,0Fp1Fp2...Fpm-1.
When A = (ai,j) is the 2x2 matrix with a11 = a12 = a12 = 1 and
a22 = 0, in brief A = ( 1 , 1 ; 1 , 0 ), then
An = ( Fn+1 , Fn ; Fn , Fn-1 ).
When A = ( 0 , 0 , 1 ; 0 , 1 , 2 ; 1 , 1 , 1 ), then
An = ( Fn-12 , Fn-1Fn , Fn2
; 2Fn-1Fn , Fn+12-Fn-1Fn , 2FnFn+1
; Fn2 , FnFn+1 , Fn+12 ).
When A = ( Fn2 , Fn+12 , Fn+22 ;
Fn+12 , Fn+22 , Fn+32 ;
Fn+22 , Fn+32 , Fn+42 ), then
det(A) = 2(-1)n+1.