login
A328284
An extension of the Jacobsthal numbers: 0, 0, 1, followed by A001045.
2
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
0,7
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
MATHEMATICA
a[n_] := If[n>3, (2^(n-3) + (-1)^n)/3, If[n == 2, 1, 0]]; (* Jean-François Alcover, Oct 16 2019 *)
KEYWORD
nonn
AUTHOR
Paul Curtz, Oct 11 2019
EXTENSIONS
Partially edited by Peter Luschny, Nov 12 2019
STATUS
approved