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

%I A328284 #49 Nov 12 2019 11:20:12
%S A328284 0,0,1,0,1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,
%T A328284 43691,87381,174763,349525,699051,1398101,2796203,5592405,11184811,
%U A328284 22369621,44739243,89478485
%N A328284 An extension of the Jacobsthal numbers: 0, 0, 1, followed by A001045.
%H A328284 OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>
%H A328284 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,2).
%F A328284 a(n) is the fourth row of the following array:
%F A328284   0,  0,  0,  0, 0, 1, 3, 7, 14, 27, 51, 97, ...
%F A328284   0,  0,  0,  0, 1, 2, 4, 7, 13, 24, 46, 89, ... = A086445
%F A328284   0,  0,  0,  1, 1, 2, 3, 6, 11, 22, 43, 86, ... = 0, 0, 0, A005578(n)
%F A328284   0,  0,  1,  0, 1, 1, 3, 5, 11, 21, 43, 85, ... = a(n)
%F A328284   0,  1, -1,  1, 0, 2, 2, 6, 10, 22, 42, 86, ...
%F A328284   1, -2,  2, -1, 2, 0, 4, 4, 12, 20, 44, 84, ...
%F A328284 From the main diagonal onward, every row is an autosequence of the first kind.
%F A328284 From _Stefano Spezia_, Oct 16 2019: (Start)
%F A328284 O.g.f.: x^2*(-1 + x + x^2)/(-1 + x + 2*x^2).
%F A328284 E.g.f.: (1/24)*exp(-x)*(8 - 9*exp(x) + exp(3*x) + 6*exp(x)*x + 6*exp(x)*x^2).
%F A328284 a(n) = a(n-1) + 2*a(n-2) for n > 4. (End)
%F A328284 a(n) = Sum_{k=0..n-1} A183190(n-k-2, n-2*k-2). - _Jean-François Alcover_, Nov 10 2019
%t A328284 a[n_] := If[n>3, (2^(n-3) + (-1)^n)/3, If[n == 2, 1, 0]]; (* _Jean-François Alcover_, Oct 16 2019 *)
%Y A328284 Cf. A000129, A001045, A054977, A139756, A141413, A166444, A183190.
%Y A328284 Cf. A000079, A005578, A063524, A086445, A139818, A257113.
%K A328284 nonn
%O A328284 0,7
%A A328284 _Paul Curtz_, Oct 11 2019
%E A328284 Partially edited by _Peter Luschny_, Nov 12 2019