The Jacobsthal sequence is an additive sequence similar to the Fibonacci sequence, defined by the recurrence relation , with initial terms and . A number in the sequence is called a Jacobsthal number. The first few are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, etc., listed in A001045 of Sloane’s OEIS.
(the denominators are powers of two). This suggests a closed form: by putting the series solution over a common denominator and summing the geometric series in the numerator, we obtain two equations, one for even-indexed terms of the sequence,
and the other one for the odd-indexed terms,
These equations can be further generalized to
The Jacobsthal numbers are named after the German mathematician Ernst Jacobsthal.
|Date of creation||2013-03-22 18:09:40|
|Last modified on||2013-03-22 18:09:40|
|Last modified by||PrimeFan (13766)|