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
)
{\displaystyle 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 the
OEIS )
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, … (sequence
A049883 in the
OEIS )
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.
{\displaystyle 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
=
2
J
n
+
(
−
1
)
n
,
{\displaystyle J_{n+1}=2J_{n}+(-1)^{n},}
or by
J
n
+
1
=
2
n
−
J
n
.
{\displaystyle 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:
[2]
J
n
=
2
n
−
(
−
1
)
n
3
.
{\displaystyle J_{n}={\frac {2^{n}-(-1)^{n}}{3}}.}
The
generating function for the Jacobsthal numbers is
x
(
1
+
x
)
(
1
−
2
x
)
.
{\displaystyle {\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
{\displaystyle J_{-n}=(-1)^{n+1}J_{n}/2^{n}}
(see
OEIS :
A077925 )
The following identity holds
2
n
(
J
−
n
+
J
n
)
=
3
J
n
2
{\displaystyle 2^{n}(J_{-n}+J_{n})=3J_{n}^{2}}
(see
OEIS :
A139818 )
Jacobsthal–Lucas numbers
Jacobsthal–Lucas numbers represent the complementary Lucas sequence
V
n
(
1
,
−
2
)
{\displaystyle V_{n}(1,-2)}
. They satisfy the same recurrence relation as Jacobsthal numbers but have different initial values:
j
n
=
{
2
if
n
=
0
;
1
if
n
=
1
;
j
n
−
1
+
2
j
n
−
2
if
n
>
1.
{\displaystyle 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:
[2]
j
n
+
1
=
2
j
n
−
3
(
−
1
)
n
.
{\displaystyle 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:
[2]
j
n
=
2
n
+
(
−
1
)
n
.
{\displaystyle 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, … (sequence
A014551 in the
OEIS ).
Jacobsthal Oblong numbers
The first Jacobsthal Oblong numbers are:
0, 1, 3, 15, 55, 231, 903, 3655, 14535, 58311, … (sequence
A084175 in the
OEIS )
J
o
n
=
J
n
J
n
+
1
{\displaystyle Jo_{n}=J_{n}J_{n+1}}
References
Possessing a specific set of other numbers
Expressible via specific sums