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Jacobsthal Number

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The Jacobsthal numbers are the numbers obtained by the U_ns in the Lucas sequence with P=1 and Q=-2, corresponding to a=2 and b=-1. They and the Jacobsthal-Lucas numbers (the V_ns) satisfy the recurrence relation

J_n=J_(n-1)+2J_(n-2)
(1)

for n>=2, with the analogous recurrence for j_n. The Jacobsthal numbers satisfy J_0=0 and J_1=1 and are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ... (OEIS A001045). The Jacobsthal-Lucas numbers satisfy j_0=2 and j_1=1 and are 2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, ... (OEIS A014551). The properties of these numbers are summarized in Horadam (1996).

Microcontrollers (and other computers) use conditional instructions to change the flow of execution of a program. In addition to branch instructions, some microcontrollers use skip instructions which conditionally bypass the next instruction. This winds up being useful for one case out of the four possibilities on 2 bits, 3 cases on 3 bits, 5 cases on 4 bits, 11 on 5 bits, 21 on 6 bits, 43 on 7 bits, 85 on 8 bits, ..., which are exactly the Jacobsthal numbers (Hirst 2006).

The Jacobsthal and Jacobsthal-Lucas numbers are given, for n>=1, by the closed form expressions

J_n=sum_(r=0)^(|_(n-1)/2_|)(n-1-r; r)2^r
(2)
j_n=sum_(r=0)^(|_n/2_|)n/(n-r)(n-r; r)2^r,
(3)

where |_x_| is the floor function and (n; k) is a binomial coefficient. The Binet forms are

J_n=1/3(a^n-b^n)
(4)
=1/3[2^n-(-1)^n]
(5)
j_n=a^n+b^n
(6)
=2^n+(-1)^n.
(7)

Amazingly, when interpreted in binary, the Jacobsthal numbers J_(n+2) give the nth iteration of applying the rule 28 cellular automaton to initial conditions consisting of a single black cell (E. W. Weisstein, pers. comm., Apr. 12, 2006).

The generating functions are

sum_(i=1)^inftyJ_ix^(i-1)=(1-x-2x^2)^(-1)
(8)
sum_(i=1)^inftyj_ix^(i-1)=(1+4x)(1-x-2x^2)^(-1).
(9)

The Simson formulas, valid for n>=1, are

J_(n+1)J_(n-1)-J_n^2=(-1)^n2^(n-1)
(10)
j_(n+1)j_(n-1)-j_n^2=9(-1)^(n-1)2^(n-1).
(11)

Summation formulas, valid for n>=1, include

sum_(i=2)^(n)J_i=1/2(J_(n+2)-3)
(12)
sum_(i=1)^(n)j_i=1/2(j_(n+2)-5).
(13)

Interrelationships are given below, with indices restricted so that all subscripts are nonnegative. In particular, n>=2 in the identity involving n-2, n>=r>=0 in those involving n-r, and m>=n>=0 in those involving m-n.

j_nJ_n=J_(2n)
(14)
j_n=J_(n+1)+2J_(n-1)
(15)
9J_n=j_(n+1)+2j_(n-1)
(16)
j_(n+1)+j_n=3(J_(n+1)+J_n)=3·2^n
(17)
j_(n+1)-j_n=3(J_(n+1)-J_n)+4(-1)^(n+1)=2^n+2(-1)^(n+1)
(18)
j_(n+1)-2j_n=3(2J_n-J_(n+1))=3(-1)^(n+1)
(19)
2j_(n+1)+j_(n-1)=3(2J_(n+1)+J_(n-1))+6(-1)^(n+1)
(20)
j_(n+r)+j_(n-r)=3(J_(n+r)+J_(n-r))+4(-1)^(n-r)
(21)
=2^(n-r)(2^(2r)+1)+2(-1)^(n-r)
(22)
j_(n+r)-j_(n-r)=3(J_(n+r)-J_(n-r))=2^(n-r)(2^(2r)-1)
(23)
j_n=3J_n+2(-1)^n
(24)
3J_n+j_n=2^(n+1)
(25)
J_n+j_n=2J_(n+1)
(26)
j_(n+2)j_(n-2)-j_n^2=-9(J_(n+2)J_(n-2)-J_n^2)=9(-1)^n2^(n-2)
(27)
J_mj_n+J_nj_m=2J_(m+n)
(28)
j_mj_n+9J_mJ_n=2j_(m+n)
(29)
j_n^2+9J_n^2=2j_(2n)
(30)
J_mj_n-J_nj_m=(-1)^n2^(n+1)J_(m-n)
(31)
j_mj_n-9J_mJ_n=(-1)^n2^(n+1)j_(m-n)
(32)
j_n^2-9J_n^2=(-1)^n2^(n+2)
(33)

(Horadam 1996).

See also

Jacobsthal-Lucas Polynomial, Jacobsthal Polynomial, Rule 28

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References

Bergum, G. E.; Bennett, L.; Horadam, A. F.; and Moore, S. D. "Jacobsthal Polynomials and a Conjecture Concerning Fibonacci-Like Matrices." Fib. Quart. 23, 240-248, 1985.Hirst, C. "Hopscotch--Multiple Bit Testing." May 15, 2006. http://www.avrfreaks.net/index.php?module=FreaksAcademy&func=viewItem&item_id=229&item_type=project.Hoggatt, V. E. Jr. and Bicknell, M. "Convolution Triangles." Fib. Quart. 10, 599-608, 1972.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. https://doi.org/10.1080/00150517.1996.12429096.Sloane, N. J. A. Sequences A001045/M2482 and A014551 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Jacobsthal Number

Cite this as:

Weisstein, Eric W. "Jacobsthal Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/JacobsthalNumber.html

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