In mathematics, the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers, they are a specific type of Lucas sequence for which P = 1, and Q = −2—and are defined by a similar recurrence relation: in simple terms, the sequence starts with 0 and 1, then each following number is found by adding the number before it to twice the number before that. The first Jacobsthal numbers are:
- 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, … (sequence A001045 in the OEIS)
Jacobsthal numbers are defined by the recurrence relation:
The next Jacobsthal number is also given by the recursion formula:
The first recursion formula above is also satisfied by the powers of 2.
The Jacobsthal number at a specific point in the sequence may be calculated directly using the closed-form equation:
The generating function for the Jacobsthal numbers is
The sum of the reciprocals of the Jacobsthal numbers is approximately 2.7186, slightly larger than e.
Jacobsthal-Lucas numbers represent the complementary Lucas sequence . They satisfy the same recurrence relation as Jacobsthal numbers but have different initial values:
The following Jacobsthal-Lucas number also satisfies:
The Jacobsthal-Lucas number at a specific point in the sequence may be calculated directly using the closed-form equation:
The first Jacobsthal-Lucas numbers are:
- 2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, 2047, 4097, 8191, 16385, 32767, 65537, 131071, 262145, 524287, 1048577, … (sequence A014551 in the OEIS).
- Weisstein, Eric W. "Jacobsthal Number". MathWorld.
- Sloane, N. J. A. (ed.). "Sequence A001045 (Jacobsthal sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- Sloane, N. J. A. (ed.). "Sequence A014551 (Jacobsthal-Lucas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.