The Jacobsthal numbers are the numbers obtained by the s in the Lucas Sequence with and ,
corresponding to and . They and the JacobsthalLucas 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, ... (Sloane's A001045).
The JacobsthalLucas 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
where
is the Floor Function and is a Binomial Coefficient. The Binet forms are
The Generating Functions are

(6) 

(7) 
The Simson Formulas are

(8) 

(9) 
Summation Formulas include
Interrelationships are

(12) 

(13) 

(14) 

(15) 

(16) 

(17) 

(18) 

(21) 

(22) 

(23) 

(24) 

(25) 

(26) 

(27) 

(28) 

(29) 

(30) 

(31) 
(Horadam 1996).
References
Horadam, A. F. ``Jacobsthal and Pell Curves.'' Fib. Quart. 26, 7983, 1988.
Horadam, A. F. ``Jacobsthal Representation Numbers.'' Fib. Quart. 34, 4054, 1996.
Sloane, N. J. A. Sequences
A014551 and
A001045/M2482
in ``An OnLine 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.
© 19969 Eric W. Weisstein
19990525