In mathematics, the Jacobsthal numbers are an

integer sequence In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. Fo ...

named after the

German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ger ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

Ernst Jacobsthal. Like the related

Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...

s, they are a specific type of

Lucas sequence In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation : x_n = P \cdot x_ - Q \cdot x_ where P and Q are fixed integers. Any sequence satisfying this r ...

U_n(P,Q)

for which ''P'' = 1, and ''Q'' = −2—and are defined by a similar

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a paramete ...

: 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 Year 171 ( CLXXI) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Herennianus (or, less frequently, year 924 ''Ab urbe co ...

, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, … A Jacobsthal prime is a Jacobsthal number that is also

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

. The first Jacobsthal primes are: :3, 5, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243, …

Jacobsthal numbers

Jacobsthal numbers are defined by the recurrence relation: :

J_n =  
    \begin
      0                     & \mbox n = 0; \\
      1                     & \mbox n = 1; \\
      J_ + 2J_    & \mbox n > 1. \\
    \end

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

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

or by: :

J_ = 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
    3.

The

generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary ser ...

for the Jacobsthal numbers is :

\frac.

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_ = (-1)^ J_n / 2^n

(see ) The following identity holds

2^n(J_ + J_n) = 3 J_n^2

(see )

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
      2                     & \mbox n = 0; \\
      1                     & \mbox n = 1; \\
      j_ + 2j_    & \mbox n > 1. \\
    \end

The following Jacobsthal–Lucas number also satisfies: :

j_ = 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,

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, 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_ = J_ J_

References

{{Classes of natural numbers Integer sequences Recurrence relations