The earliest use of the phrase “Jacobsthal numbers” I could find was in another paper by Horadam, “Jacobsthal and Pell Curves,” in The Fibonacci Quarterly 26 (1988), 77-83. This paper also cites the original paper by Jacobsthal. (The Jacobsthal paper is also cited in a paper by Bergum, Bennett, Horadam and Moore, “Jacobsthal Polynomials and a Conjecture Concerning Fibonacci-like Matrices,” in FQ 23 (1985), 240-248, with a nice historical note.)

Prior to the Horadam paper, there are numerous appearances of the phrase “Jacobsthal polynomials” (the earliest I could find is by Hoggatt and Bicknell, in “Convolution Triangles,” FQ 10 (1972), 599-608), but none of the papers I’ve looked at is clearly the Ur-spring for the terminology. The sequence 1,3,5,11,21,… even shows up in some of these papers, but is not singled out or named. Curiously, the sequence is explicitly studied, but not named (nor Jacobsthal even mentioned), in another FQ (1972) paper by Hoggatt and two other co-authors, one issue earlier than the Hoggatt/Bicknell paper. The first appearance of the sequence in connection with Jacobsthal polynomials is in an FQ (1978) paper by Hoggatt and Bicknell (by then Bicknell-Johnson), where it’s part of an array of numbers.

Unfortunately FQ is not part of JSTOR (although it does have its own online index), so it’s hard to be sure I haven’t missed something. (JSTOR has nothing to offer under either “Jacobsthal number” or “Jacobsthal polynomial.” MathSciNet returns nothing earlier than 1978.) I’m afraid this is the best I can do for now.