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%I #17 Sep 23 2019 15:56:00
%S 0,1,1,2,4,8,17,36,77,165,354,760,1632,3505,7528,16169,34729,74594,
%T 160220,344136,739169,1587660,3410133,7324621,15732546,33791920,
%U 72581632,155898017,334853200,719230865,1544835281,3318150210,7127051636,15308187336
%N Expansion of x*(1-x-x^2)/((1-x)*(1-x-2*x^2-x^3)).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-1,-1).
%F G.f.: A(x) = x*(1-x-x^2)*B(x), where B is g.f. of A077864.
%F a(n) = A077864(n+1)-2*A077864(n), n >= 0.
%F a(n+3) = A077864(n+2)-A077864(n+1)-A077864(n), n >= 0.
%F Recurrence: a(0)=0, a(1)=1, a(2)=1, a(3)=2, a(4)=4, a(5)=8, a(6)=17, and a(n) = 4*a(n-1)-4*a(n-2)+a(n-7) for n >= 7.
%F Conjecture: a(n+1) = Sum_{j=0..n/2} A027907(n-j,2*j), n >= 0.
%F a(n) = 2*a(n-1)+a(n-2)-a(n-3)-a(n-4) for n>3. - _Wesley Ivan Hurt_, Oct 10 2015
%F a(n) = a(n-1)+2*a(n-2)+a(n-3)-1, n>=3. - _R. J. Mathar_, Nov 07 2015
%p a:=proc(n) option remember; if n=0 then 0 elif n=1 then 1 elif n=2 then 1 elif n=3 then 2 else 2*a(n-1)+a(n-2)-a(n-3)-a(n-4); fi; end: seq(f(n), n=0..50); # _Wesley Ivan Hurt_, Oct 10 2015
%t CoefficientList[Series[x (1 - x - x^2)/((1 - x) (1 - x - 2 x^2 - x^3)), {x, 0, 100}], x] (* _Vincenzo Librandi_, Sep 29 2015 *)
%t LinearRecurrence[{2,1,-1,-1},{0,1,1,2},40] (* _Harvey P. Dale_, Sep 23 2019 *)
%o (PARI) concat(0, Vec(x*(1-x-x^2)/((1-x)*(1-x-2*x^2-x^3)) + O(x^50))) \\ _Michel Marcus_, Sep 29 2015
%Y Cf. A027907, A077864.
%K nonn,easy
%O 0,4
%A _Werner Schulte_, Sep 29 2015
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