Jacobsthal number

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 second 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:

:J_n = \frac{2^n - (-1)^n}{3}. The generating function for the Jacobsthal numbers is

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

The sum of the reciprocals of the Jacobsthal numbers is approximately 2.7186, slightly larger than e.

The Jacobsthal numbers can be extended to negative indices using the recurrence relation or the explicit formula, giving

:J_{-n} = (-1)^{n+1} J_n / 2^n (see )

The following identities holds

:2^n(J_{-n} + J_n) = 3 J_n^2 (see )

:J_n = F_n + \sum_{i=0}^{n-2}J_iF_{n-i-1} where F_n is the nth Fibonacci number.

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:

: j_n = \begin{cases} 2 & \mbox{if } n = 0; \ 1 & \mbox{if } n = 1; \ j_{n-1} + 2j_{n-2} & \mbox{if } n 1. \ \end{cases}

The following Jacobsthal–Lucas number also satisfies:

: j_{n+1} = 2j_n - 3(-1)^n. ,

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

: j_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, … .

Jacobsthal Oblong numbers

The first Jacobsthal Oblong numbers are: 0, 1, 3, 15, 55, 231, 903, 3655, 14535, 58311, …

:Jo_{n} = J_{n} J_{n+1}

References

References

1. "Jacobsthal Number".
2. {{Cite OEIS