Lehmer sequence

In mathematics, a Lehmer sequence $U_{n}({\sqrt {R}},Q)$ or $V_{n}({\sqrt {R}},Q)$ is a generalization of a Lucas sequence $U_{n}(P,Q)$ or $V_{n}(P,Q)$ , allowing the square root of an integer R in place of the integer P.^[1]

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:

{\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={\sqrt {R}}

ab=Q

under the following conditions:

Q and R are relatively prime nonzero integers
$a/b$ is not a root of unity.

Then, the corresponding Lehmer numbers are:

U_{n}({\sqrt {R}},Q)={\frac {a^{n}-b^{n}}{a-b}}

for n odd, and

U_{n}({\sqrt {R}},Q)={\frac {a^{n}-b^{n}}{a^{2}-b^{2}}}

for n even.

Their companion numbers are:

V_{n}({\sqrt {R}},Q)={\frac {a^{n}+b^{n}}{a+b}}

for n odd and

V_{n}({\sqrt {R}},Q)=a^{n}+b^{n}

for n even.

Recurrence

Lehmer numbers form a linear recurrence relation with

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 $U_{0}=0,\,U_{1}=1,\,U_{2}=1,\,U_{3}=R-Q=a^{2}+ab+b^{2}$ . Similarly the companion sequence satisfies

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 $V_{0}=2,\,V_{1}=1,\,V_{2}=R-2Q=a^{2}+b^{2},\,V_{3}=R-3Q=a^{2}-ab+b^{2}.$

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. For example,

{\begin{aligned}U_{2n}({\sqrt {R}},Q)&={\phantom {R\,}}U_{2n-1}({\sqrt {R}},Q)-Q\,U_{2n-2}({\sqrt {R}},Q)&U_{2n+1}({\sqrt {R}},Q)&=R\,U_{2n}({\sqrt {R}},Q)-Q\,U_{2n-1}({\sqrt {R}},Q)\\V_{2n}({\sqrt {R}},Q)&=R\,V_{2n-1}({\sqrt {R}},Q)-Q\,V_{2n-2}({\sqrt {R}},Q)&V_{2n+1}({\sqrt {R}},Q)&={\phantom {R\,}}V_{2n}({\sqrt {R}},Q)-Q\,V_{2n-1}({\sqrt {R}},Q)\end{aligned}}

References

^ Weisstein, Eric W. "Lehmer Number". mathworld.wolfram.com. Retrieved 2020-08-11.

[1] Weisstein, Eric W. "Lehmer Number". mathworld.wolfram.com. Retrieved 2020-08-11.

[1]

Fahrer / Team für die Suche eingeben:

Fahrer / Team für die Suche eingeben:

Langbahn Team – Weltmeisterschaft

Lehmer sequence

Algebraic relations

Recurrence

References