The Jacobsthal numbers follow the recurrence \(J_n = J_{n-1} + 2J_{n-2}\) from seeds \(0, 1\): they run 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365. The doubling of the second term (instead of Fibonacci's plain sum) gives them a neat closed form, \(J_n = (2^n - (-1)^n)/3\), so consecutive Jacobsthal numbers hug the powers of two.
They show up in tiling and domino problems, in the number of ways to tile a strip, and in certain set-partition counts. Named after the German mathematician Ernst Jacobsthal. The associated "Jacobsthal-Lucas" companion sequence relates to them just as the Lucas numbers relate to Fibonacci.