Lucas Numbers

The Lucas numbers are an integer sequence defined by the same recurrence relation as the Fibonacci sequence, in which each term is the sum of the two preceding terms, but beginning from the different starting values two and one rather than zero and one, producing the sequence two, one, three, four, seven, eleven, eighteen, twenty-nine and so on. The sequence is named after the French mathematician Édouard Lucas, who studied it extensively in the nineteenth century alongside the closely related Fibonacci sequence, and who also devised primality tests building on both sequences that are still used, in refined form, to test whether certain large numbers, including Mersenne numbers, are prime. The Lucas and Fibonacci sequences are linked by simple identities, for instance a Lucas number equals the sum of the Fibonacci numbers immediately before and after the corresponding index, and both sequences share the same limiting ratio of consecutive terms, the golden ratio.