Jacobsthal number
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[1]—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:
Jacobsthal numbers
Jacobsthal numbers are defined by the recurrence relation:
The next Jacobsthal number is also given by the recursion formula:
or by:
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:[2]
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
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:[3]
The Jacobsthal-Lucas number at a specific point in the sequence may be calculated directly using the closed-form equation:[3]
The first Jacobsthal-Lucas numbers are:
References
- ↑ Weisstein, Eric W. "Jacobsthal Number". MathWorld.
- ↑ Sloane, N.J.A. (ed.). "Sequence A001045 (Jacobsthal sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 Sloane, N.J.A. (ed.). "Sequence A014551 (Jacobsthal-Lucas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.