FT13: Summary of Formulas for F(kn+m) and L(kn+m)

The previous two sections covered a variety of formulas for F(kn+m) and L(kn+m); we summarize them here.

Formulas for the Fibonacci numbers:

• F (kn + m) = ∑_{0 ≤ j ≤ k} (  k   j  ) F ( j + m ) F(n) ^j F(n−1)^k−j, for any k.

• 5^k/2 F (kn + m) = ∑_{0 ≤ j ≤ k} (  k   j  ) F ( j + m ) L(n) ^j L(n−1)^k−j, for k even.

• 5^(k+1)/2 F(kn+m)
  = ∑_{0 ≤ j ≤ k} (  k   j  ) L( j+m ) L(n) ^j L(n−1)^k−j, for k odd.

Formulas for the Lucas numbers:

• L (kn + m) = ∑_{0 ≤ j ≤ k} (  k   j  ) L ( j + m ) F(n) ^j F(n−1)^k−j, for any k.

• 5^k/2 L (kn + m) = ∑_{0 ≤ j ≤ k} (  k   j  ) L ( j + m ) L(n) ^j L(n−1)^k−j, for k even.

• 5^(k−1)/2 L(kn+m)
  = ∑_{0 ≤ j ≤ k} (  k   j  ) F( j+m ) L(n) ^j L(n−1)^k−j, for k odd.