Pensieve header: A formula for the Fibonacci numbers, plus some tidbits.

{(Subscript[λ, 1]),(Subscript[λ, 2])} = x /.Solve[(x^2)-x-1==0]

F[n_] := Expand[a ((Subscript[λ, 1])^n)+b ((Subscript[λ, 2])^n)];
Solve[F[0]==1∧F[1]==1, {a,b}] /. Rule -> Set

F[n] // Simplify

F[1000]

Table[F[n], {n, 0, 10}]

{F[21]/F[20], GoldenRatio} // N

n=30; F[n]

Homework (not to be submitted). What is the story told by each of the following formulas?

{((F[(n/2)])^2)+((F[(n/2)-1])^2), (∑_(k=0)^(n/2))Binomial[n-k,k], (∑_(k=0)^(n-2))F[k], (∑_(k=0)^(n/2-1))F[n-2k-1]}

Table[Binomial[n,k], {n,0,10}, {k,0,10}] // MatrixForm