Jacobsthal number - Alchetron, The Free Social Encyclopedia

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—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:

J n = { 0 if n = 0 ; 1 if n = 1 ; J n − 1 + 2 J n − 2 if n > 1.

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

J n + 1 = 2 J 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:

J n = 2 n − ( − 1 ) n 3 .

The generating function for the Jacobsthal numbers is

x ( 1 + x ) ( 1 − 2 x ) .

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 = { 2 if n = 0 ; 1 if n = 1 ; L n − 1 + 2 L n − 2 if n > 1.

The following Jacobsthal-Lucas number also satisfies:

L n + 1 = 2 L n − 3 ( − 1 ) n .

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

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 the OEIS).

References

Jacobsthal number Wikipedia

(Text) CC BY-SA

Contents

Jacobsthal numbers

Jacobsthal-Lucas numbers

References