Jacobsthal Number

The Jacobsthal numbers are the numbers obtained by the s in the Lucas Sequence with and , corresponding to and . They and the Jacobsthal-Lucas numbers (the s) satisfy the Recurrence Relation

$\begin{displaymath} J_n=J_{n-1}+2J_{n-2}. \end{displaymath}$

(1)

The Jacobsthal numbers satisfy

and

and are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ... (Sloane's A001045). The Jacobsthal-Lucas numbers satisfy

and

and are 2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, ... (Sloane's A014551). The properties of these numbers are summarized in Horadam (1996). They are given by the closed form expressions

$\displaystyle J_n$	$\textstyle =$	$\displaystyle \sum_{r=0}^{\left\lfloor{(n-1)/2}\right\rfloor } {n-1-r\choose r}2^r$	(2)
$\displaystyle j_n$	$\textstyle =$	$\displaystyle \sum_{r=0}^{\left\lfloor{n/2}\right\rfloor } {n\over n-r}{n-r\choose r}2^r,$	(3)

where $\left\lfloor{x}\right\rfloor$ is the Floor Function and ${n\choose k}$ is a Binomial Coefficient. The Binet forms are

$\displaystyle J_n$	$\textstyle =$	$\displaystyle {\textstyle{1\over 3}}(a^n-b^n)={\textstyle{1\over 3}}[2^n-(-1)^n]$	(4)
$\displaystyle j_n$	$\textstyle =$	$\displaystyle a^n+b^n=2^n+(-1)^n.$	(5)

The Generating Functions are

$\begin{displaymath} \sum_{i=1}^\infty J_ix^{i-1}=(1-x-2x^2)^{-1} \end{displaymath}$

(6)

$\begin{displaymath} \sum_{i=1}^\infty j_ix^{i-1}=(1+4x)(1-x-2x^2)^{-1}. \end{displaymath}$

(7)

The Simson Formulas are

$\begin{displaymath} J_{n+1}J_{n-1}-{J_n}^2=(-1)^n2^{n-1} \end{displaymath}$

(8)

$\begin{displaymath} j_{n+1}j_{n-1}-{j_n}^2=9(-1)^{n-1}2^{n-1}=-9(J_{n+1}J_{n-1}-{J_n}^2). \end{displaymath}$

(9)

Summation Formulas include

$\displaystyle \sum_{i=2}^n J_i$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}(J_{n+2}-3)$	(10)
$\displaystyle \sum_{i=1}^n j_i$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}(j_{n+2}-5).$	(11)

Interrelationships are

$\begin{displaymath} j_nJ_n=J_{2n} \end{displaymath}$

(12)

$\begin{displaymath} j_n=J_{n+1}+2J_{n-1} \end{displaymath}$

(13)

$\begin{displaymath} 9J_n=j_{n+1}+2j_{n-1} \end{displaymath}$

(14)

$\begin{displaymath} j_{n+1}+j_n=3(J_{n+1}+J_n)=3\cdot 2^n \end{displaymath}$

(15)

$\begin{displaymath} j_{n+1}-j_n=3(J_{n+1}-J_n)+4(-1)^{n+1}=2^n+2(-1)^{n+1} \end{displaymath}$

(16)

$\begin{displaymath} j_{n+1}-2j_n=3(2J_n-J_{n+1})=3(-1)^{n+1} \end{displaymath}$

(17)

$\begin{displaymath} 2j_{n+1}+j_{n-1}=3(2J_{n+1}+J_{n-1})+6(-1)^{n+1} \end{displaymath}$

(18)

$\displaystyle j_{n+r}+j_{n-r}$	$\textstyle =$	$\displaystyle 3(J_{n+r}+J_{n-r})+4(-1)^{n-r}$	(19)
	$\textstyle =$	$\displaystyle 2^{n-r}(2^{2r}+1)+2(-1)^{n-r}$	(20)

$\begin{displaymath} j_{n+r}-j_{n-r}=3(J_{n+r}-J_{n-r})=2^{n-r}(2^{2r}-1) \end{displaymath}$

(21)

$\begin{displaymath} j_n=3J_n+2(-1)^n \end{displaymath}$

(22)

$\begin{displaymath} 3J_n+j_n=2^{n+1} \end{displaymath}$

(23)

$\begin{displaymath} J_n+j_n=2J_{n+1} \end{displaymath}$

(24)

$\begin{displaymath} j_{n+2}j_{n-2}-{j_n}^2=-9(J_{n+2}J_{n-2}-{J_n})^2=9(-1)^n 2^{n-2} \end{displaymath}$

(25)

$\begin{displaymath} J_mj_n+J_nj_m=2J_{m+n} \end{displaymath}$

(26)

$\begin{displaymath} j_mj_n+9J_mJ_n=2j_{m+n} \end{displaymath}$

(27)

$\begin{displaymath} {j_n}^2+9{J_n}^2=2j_{2n} \end{displaymath}$

(28)

$\begin{displaymath} J_mj_n-J_nj_m=(-1)^n2^{n+1}J_{m-n} \end{displaymath}$

(29)

$\begin{displaymath} j_mj_n-9J_mJ_n=(-1)^n2^{n+1}j_{m-n} \end{displaymath}$

(30)

$\begin{displaymath} {j_n}^2-9{J_n}^2=(-1)^n2^{n+2} \end{displaymath}$

(31)

(Horadam 1996).

References

Horadam, A. F. ``Jacobsthal and Pell Curves.'' Fib. Quart. 26, 79-83, 1988.

Horadam, A. F. ``Jacobsthal Representation Numbers.'' Fib. Quart. 34, 40-54, 1996.

Sloane, N. J. A. Sequences A014551 and A001045/M2482 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.