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A238824 a(1)=0, a(2)=1; thereafter a(n) = A238823(n-1)-2*a(n-1)-a(n-2).

%I A238824 #18 Dec 26 2023 13:24:05
%S A238824 0,1,1,3,7,17,43,105,262,643,1590,3911,9643,23743,58502,144099,355009,
%T A238824 874545,2154505,5307663,13075683,32212375,79356454,195497421,
%U A238824 481615082,1186475969,2922926441,7200734309,17739268544,43701326725,107659793177,265223778927
%N A238824 a(1)=0, a(2)=1; thereafter a(n) = A238823(n-1)-2*a(n-1)-a(n-2).
%H A238824 Vincenzo Librandi, <a href="/A238824/b238824.txt">Table of n, a(n) for n = 1..1000</a>
%H A238824 V. M. Zhuravlev, <a href="http://www.mccme.ru/free-books/matpros/mph.pdf">Horizontally-convex polyiamonds and their generating functions</a>, Mat. Pros. 17 (2013), 107-129 (in Russian). See the sequence d(n).
%H A238824 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-4,-3,2,4,2,-1).
%F A238824 G.f.: -x^2*(-1+x+2*x^2-2*x^3-x^4+x^6-x^5) / ( (1+x)*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1) ). - _R. J. Mathar_, Mar 20 2014
%p A238824 g:=proc(n) option remember; local t1;
%p A238824 t1:=[2,3,6,14,34,84,208,515];
%p A238824 if n <= 7 then t1[n] else
%p A238824 3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc;
%p A238824 [seq(g(n),n=1..32)]; # A238823
%p A238824 d:=proc(n) option remember; global g; local t1;
%p A238824 t1:=[0,1];
%p A238824 if n <= 2 then t1[n] else
%p A238824 g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
%p A238824 [seq(d(n),n=1..32)]; # A238824
%t A238824 CoefficientList[Series[-x (- 1 + x + 2 x^2 - 2 x^3 - x^4 + x^6 - x^5)/((1 + x) (x^7 - 3 x^6 - x^5 - x^4 + 4 x^3 - 3 x + 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 21 2014 *)
%t A238824 LinearRecurrence[{2,3,-4,-3,2,4,2,-1},{0,1,1,3,7,17,43,105},40] (* _Harvey P. Dale_, Dec 26 2023 *)
%o A238824 (Magma) m:=40; R<x>:=LaurentSeriesRing(RationalField(), m); [0] cat Coefficients(R! -x^2*(-1+x+2*x^2-2*x^3-x^4+x^6-x^5) / ( (1+x)*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1))); // _Vincenzo Librandi_, Mar 21 2014
%K A238824 nonn,easy
%O A238824 1,4
%A A238824 _N. J. A. Sloane_, Mar 08 2014