07 What is the Lucas Sequence?

A000032 – OEIS

The (special) Lucas sequence is named after the French mathematician Édouard Lucas, who was the first to study them.

The sequence is calculated recursively with

Alternatively, the closed formula can be used:

The first numbers of the special Lucas sequence are
2,
1,
3,
4,
7,
11,
18,
29,
47,
76,
123,
199,
322,
521,
843,
1364,
2207,
3571,
5778,
9349,
15127,
24476,
39603,
64079,
103682,
167761,
271443,
439204,
710647,
1149851,
1860498,
3010349,
4870847,
7881196,
12752043,
20633239,
33385282,
54018521,
87403803

General Lucas sequence

Lucas also discovered formation formulas for the general Lucas sequences Un(P, Q) and Vn(P, Q), which, given appropriate choices of P and Q, lead to other known number sequences:

PQU(P, Q)V(P,Q)
1-1Fibonacci SequenceLucas Sequence
1-2Jacobsthal SequenceJacobsthal-Lucas Sequence
2-1Pell SequencePell-Lucas Sequence
32Mersenne SequenceMersenne-like Sequence including the Fermat Numbers