Un(P,Q)
0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, …
A Jacobsthal prime is a Jacobsthal number that is also prime. The first Jacobsthal primes are:
3, 5, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243, …
Jacobsthal numbers are defined by the recurrence relation:
Jn=\begin{cases} 0&ifn=0;\\ 1&ifn=1;\\ Jn-1+2Jn-2&ifn>1.\\ \end{cases}
The next Jacobsthal number is also given by the recursion formula
Jn+1=2Jn+(-1)n,
or by
Jn+1=2n-Jn.
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:
Jn=
2n-(-1)n | |
3 |
.
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+1Jn/2n
The following identity holds
n(J | |
2 | |
-n |
+Jn)=3
2 | |
J | |
n |
Jacobsthal–Lucas numbers represent the complementary Lucas sequence
Vn(1,-2)
jn=\begin{cases} 2&ifn=0;\\ 1&ifn=1;\\ jn-1+2jn-2&ifn>1.\\ \end{cases}
The following Jacobsthal–Lucas number also satisfies:[1]
jn+1=2jn-3(-1)n.
The Jacobsthal–Lucas number at a specific point in the sequence may be calculated directly using the closed-form equation:
jn=2n+(-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, … .
The first Jacobsthal Oblong numbers are: 0, 1, 3, 15, 55, 231, 903, 3655, 14535, 58311, …
Jon=JnJn+1