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

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

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),