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 \( U_n(P,Q) \) 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:

0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, … (sequence A001045 in OEIS)

Jacobsthal numbers

Jacobsthal numbers are defined by the recurrence relation:

\( J_n = \begin{cases} 0 & \mbox{if } n = 0; \\ 1 & \mbox{if } n = 1; \\ J_{n-1} + 2J_{n-2} & \mbox{if } n > 1. \\ \end{cases} \)

The next Jacobsthal number is also given by the recursion formula:

\( J_{n+1} = 2J_n + (-1)^n \, , \)

or by:

\( J_{n+1} = 2^n - J_n. \, \)

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]

\( J_n = \frac{2^n - (-1)^n} 3. \)

The generating function for the Jacobsthal numbers is

\( \frac{x}{(1+x)(1-2x)}. \)

Jacobsthal-Lucas numbers

Jacobsthal-Lucas numbers represent the complementary Lucas sequence \( V_n(1,-2) \). They satisfy the same recurrence relation as Jacobsthal numbers but have different initial values:

\( L_n = \begin{cases} 2 & \mbox{if } n = 0; \\ 1 & \mbox{if } n = 1; \\ L_{n-1} + 2L_{n-2} & \mbox{if } n > 1. \\ \end{cases} \)

The following Jacobsthal-Lucas number also satisfies:[3]

\( L_{n+1} = 2L_n - 3(-1)^n. \, \)

The Jacobsthal-Lucas number at a specific point in the sequence may be calculated directly using the closed-form equation:[3]

\( L_n = 2^n + (-1)^n. \, \)

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 OEIS).

References

Weisstein, Eric W., "Jacobsthal Number", MathWorld.
"Sloane's A001045 : Jacobsthal sequence", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
"Sloane's A014551 : Jacobsthal-Lucas numbers", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

Mathematics Encyclopedia