A328284 An extension of the Jacobsthal numbers: 0, 0, 1, followed by A001045.

0, 0, 1, 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, 44739243, 89478485

Offset: 0

Author

Paul Curtz, Oct 11 2019

Keywords

• nonn

Links

• OEIS Wiki, Autosequence
• Index entries for linear recurrences with constant coefficients, signature (1,2).

Crossrefs

Cf. A000129, A001045, A054977, A139756, A141413, A166444, A183190.

Cf. A000079, A005578, A063524, A086445, A139818, A257113.

Programs

Mathematica

a[n_] := If[n>3, (2^(n-3) + (-1)^n)/3, If[n == 2, 1, 0]]; (* Jean-François Alcover, Oct 16 2019 *)

Formula

a(n) is the fourth row of the following array:

0, 0, 0, 0, 0, 1, 3, 7, 14, 27, 51, 97, ...

0, 0, 0, 0, 1, 2, 4, 7, 13, 24, 46, 89, ... = A086445

0, 0, 0, 1, 1, 2, 3, 6, 11, 22, 43, 86, ... = 0, 0, 0, A005578(n)

0, 0, 1, 0, 1, 1, 3, 5, 11, 21, 43, 85, ... = a(n)

0, 1, -1, 1, 0, 2, 2, 6, 10, 22, 42, 86, ...

1, -2, 2, -1, 2, 0, 4, 4, 12, 20, 44, 84, ...

From the main diagonal onward, every row is an autosequence of the first kind.

From Stefano Spezia, Oct 16 2019: (Start)

O.g.f.: x^2*(-1 + x + x^2)/(-1 + x + 2*x^2).

E.g.f.: (1/24)*exp(-x)*(8 - 9*exp(x) + exp(3*x) + 6*exp(x)*x + 6*exp(x)*x^2).

a(n) = a(n-1) + 2*a(n-2) for n > 4. (End)

a(n) = Sum_{k=0..n-1} A183190(n-k-2, n-2*k-2). - Jean-François Alcover, Nov 10 2019

Extensions

Partially edited by Peter Luschny, Nov 12 2019