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# 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

 (1)

The Jacobsthal numbers satisfy and and are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ... (OEIS A001045). The Jacobsthal-Lucas numbers satisfy and 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 by the closed form expressions

 (2) (3)

where is the floor function and is a binomial coefficient. The Binet forms are

 (4) (5) (6) (7)

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

The generating functions are

 (8)
 (9)

The Simson formulas are

 (10) (11)

Summation formulas include

 (12) (13)

Interrelationships are

 (14)
 (15)
 (16)
 (17)
 (18)
 (19)
 (20)
 (21) (22)
 (23)
 (24)
 (25)
 (26)
 (27)
 (28)
 (29)
 (30)
 (31)
 (32)
 (33)

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.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 A001045/M2482 and A014551 in "The On-Line Encyclopedia of Integer Sequences."Hoggatt and Bicknell, in ÒConvolution Triangles,Ó FQ 10 (1972), 599-608),

## Referenced on Wolfram|Alpha

Jacobsthal Number

## Cite this as:

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