16.7.5 Lucas Numbers

mpz_lucnum2_ui derives a pair of Lucas numbers from a pair of Fibonacci numbers with the following simple formulas.

L[k]   =   F[k] + 2*F[k-1]
L[k-1] = 2*F[k] -   F[k-1]

mpz_lucnum_ui is only interested in L[n], and some work can be saved. Trailing zero bits on n can be handled with a single square each.

L[2k] = L[k]^2 - 2*(-1)^k

And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci numbers, similar to what mpz_fib_ui does.

L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k