Generalised Jacobsthal numbers.

Ernst Jacobsthal
(1882 - 1965)

The number associated with my diurnal age today, 27305, has the property that it is a member of OEIS A084640:

A084640

Generalised Jacobsthal numbers.

There's not a great deal of information given about Jacobsthal numbers of any sort so I did some investigation which yielded the following information.

Fibonacci Numbers:$$F_n=F_{n-1}+F_{n-2}\\ \text{where }n \geq 2, F_0=0, F_1=1$$The first 25 members of this sequence are:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393

Lucus Number:$$L_n=L_{n-1}+L_{n-2} \\ \text{where } n \geq 2, L_0=2, L_1=1$$

The first 25 members of this sequence are:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443

Pell Numbers:$$P_n=2P_{n-1}+P_{n-2} \\ \text{where } n \geq 2, P_0=0, P_1=1 $$The first 25 members of this sequence are:

0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210, 6625109, 15994428, 38613965, 93222358, 225058681, 543339720, 1311738121, 3166815962

Pell-Lucus Numbers:$$Q_n=2Q_{n-1}+Q_{n-2} \\ \text{where } n \geq 2, Q_0=Q_1=1$$The first 25 members of this sequence are:

1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243, 275807, 665857, 1607521, 3880899, 9369319, 22619537, 54608393, 131836323, 318281039, 768398401, 1855077841, 4478554083

Jacobsthal Numbers:$$J_n=J_{n-1}+2J_{n-2} \\ \text{where } n \geq 2, J_0=0, J_1=1 $$The first 25 members of this sequence are:

0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621

Jacobsthal-Lucas Numbers:$$ j_n=j_{n-1}+2j_{n-2} \\ \text{where } n \geq 2, j_0=j_1=2$$The first 25 members of this sequence are:

2, 2, 6, 10, 22, 42, 86, 170, 342, 682, 1366, 2730, 5462, 10922, 21846, 43690, 87382, 174762, 349526, 699050, 1398102, 2796202, 5592406, 11184810, 22369622, 44739242, 89478486

Thus the generalised Jacobsthal numbers referenced in OEIS A084640 arise by simply adding a constant to the formula for the Jacobsthal numbers shown earlier. The formula thus becomes:

An Example of Generalised Jacobsthal Numbers:$$J^{*}_n=J^{*}_{n-1}+J^{*}_{n-2}+4 \\ \text{where } n \geq 2, J^{*}_0=0, J^{*}_1=1$$The first 25 members of this sequence are (permalink):

0, 1, 5, 11, 25, 51, 105, 211, 425, 851, 1705, 3411, 6825, 13651, 27305, 54611, 109225, 218451, 436905, 873811, 1747625, 3495251, 6990505, 13981011, 27962025, 55924051, 111848105

The generating function is given by:$$ \frac{x(1+3x)}{(1-x^2)(1-2x)}$$The \(n\)-th term is given by:$$ \frac{(5 \times 2^n + (-1)^n - 6)}{3}$$All this doesn't tell us anything about the mathematician Jacobsthal and there's an interesting remark by Donald Knuth in the OEIS comments:

Don Knuth points out (personal communication) that Jacobsthal may never have seen the actual values of his sequence. However, Horadam uses the name "Jacobsthal sequence", such an important sequence needs a name, and there is a law that says the name for something should never be that of its discoverer. 

N. J. A. Sloane, Dec 26 2020

There is a biography of Ernst Jacobsthal to be found at MacTutor.