Lehmer sequence
Open-knowledge reference entry
In mathematics, a Lehmer sequence
U
n
(
R
,
Q
)
{\displaystyle U_{n}({\sqrt {R}},Q)}
or
V
n
(
R
,
Q
)
{\displaystyle V_{n}({\sqrt {R}},Q)}
is a generalization of a Lucas sequence
U
n
(
P
,
Q
)
{\displaystyle U_{n}(P,Q)}
or
V
n
(
P
,
Q
)
{\displaystyle V_{n}(P,Q)}
, allowing the square root of an integer R in place of the integer P.
To ensure that the value is always an integer, every other term of a Lehmer sequence is divided by √R compared to the corresponding Lucas sequence. That is, when R = P2 the Lehmer and Lucas sequences are related as:
P
U
2
n
(
P
2
,
Q
)
=
U
2
n
(
P
,
Q
)
U
2
n
+
1
(
P
2
,
Q
)
=
U
2
n
+
1
(
P
,
Q
)
V
2
n
(
P
2
,
Q
)
=
V
2
n
(
P
,
Q
)
P
V
2
n
+
1
(
P
2
,
Q
)
=
V
2
n
+
1
(
P
,
Q
)
{\displaystyle {\begin{aligned}P\,U_{2n}({\sqrt {P^{2}}},Q)&=U_{2n}(P,Q)&U_{2n+1}({\sqrt {P^{2}}},Q)&=U_{2n+1}(P,Q)\\V_{2n}({\sqrt {P^{2}}},Q)&=V_{2n}(P,Q)&P\,V_{2n+1}({\sqrt {P^{2}}},Q)&=V_{2n+1}(P,Q)\end{aligned}}}
Algebraic relations
If a and b are complex numbers with
a
+
b
=
R
{\displaystyle a+b={\sqrt {R}}}
a
b
=
Q
{\displaystyle ab=Q}
under the following conditions:
Q and R are relatively prime nonzero integers
a
/
b
{\displaystyle a/b}
is not a root of unity.
Then, the corresponding Lehmer numbers are:
U
n
(
R
,
Q
)
=
a
n
−
b
n
a
−
b
{\displaystyle U_{n}({\sqrt {R}},Q)={\frac {a^{n}-b^{n}}{a-b}}}
for n odd, and
U
n
(
R
,
Q
)
=
a
n
−
b
n
a
2
−
b
2
{\displaystyle U_{n}({\sqrt {R}},Q)={\frac {a^{n}-b^{n}}{a^{2}-b^{2}}}}
for n even.
Their companion numbers are:
V
n
(
R
,
Q
)
=
a
n
+
b
n
a
+
b
{\displaystyle V_{n}({\sqrt {R}},Q)={\frac {a^{n}+b^{n}}{a+b}}}
for n odd and
V
n
(
R
,
Q
)
=
a
n
+
b
n
{\displaystyle V_{n}({\sqrt {R}},Q)=a^{n}+b^{n}}
for n even.
Recurrence
Lehmer numbers form a linear recurrence relation with
U
n
=
(
R
−
2
Q
)
U
n
−
2
−
Q
2
U
n
−
4
=
(
a
2
+
b
2
)
U
n
−
2
−
a
2
b
2
U
n
−
4
{\displaystyle U_{n}=(R-2Q)U_{n-2}-Q^{2}U_{n-4}=(a^{2}+b^{2})U_{n-2}-a^{2}b^{2}U_{n-4}}
with initial values
U
0
=
0
,
U
1
=
1
,
U
2
=
1
,
U
3
=
R
−
Q
=
a
2
+
a
b
+
b
2
{\displaystyle U_{0}=0,\,U_{1}=1,\,U_{2}=1,\,U_{3}=R-Q=a^{2}+ab+b^{2}}
. Similarly the companion sequence satisfies
V
n
=
(
R
−
2
Q
)
V
n
−
2
−
Q
2
V
n
−
4
=
(
a
2
+
b
2
)
V
n
−
2
−
a
2
b
2
V
n
−
4
{\displaystyle V_{n}=(R-2Q)V_{n-2}-Q^{2}V_{n-4}=(a^{2}+b^{2})V_{n-2}-a^{2}b^{2}V_{n-4}}
with initial values
V
0
=
2
,
V
1
=
1
,
V
2
=
R
−
2
Q
=
a
2
+
b
2
,
V
3
=
R
−
3
Q
=
a
2
−
a
b
+
b
2
.
{\displaystyle V_{0}=2,\,V_{1}=1,\,V_{2}=R-2Q=a^{2}+b^{2},\,V_{3}=R-3Q=a^{2}-ab+b^{2}.}
All Lucas sequence recurrences apply to Lehmer sequences if they are divided into cases for even and odd n and appropriate factors of √R are incorporated.
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