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:
| P | Q | U(P, Q) | V(P,Q) |
| 1 | -1 | Fibonacci Sequence | Lucas Sequence |
| 1 | -2 | Jacobsthal Sequence | Jacobsthal-Lucas Sequence |
| 2 | -1 | Pell Sequence | Pell-Lucas Sequence |
| 3 | 2 | Mersenne Sequence | Mersenne-like Sequence including the Fermat Numbers |