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A238831 a(n) = 0 if n <= 2; thereafter a(n) = A238827(n) + A238830(n-2).

%I A238831 #19 Feb 09 2024 17:21:48
%S A238831 0,0,0,0,0,1,3,8,21,49,124,295,735,1789,4428,10874,26836,66062,162838,
%T A238831 401081,988225,2434388,5997403,14774547,36397880,89667011,220898267,
%U A238831 544190131,1340632638,3302695932,8136311688,20044096016,49379354928,121647818677,299683787423,738281805364,1818783831517
%N A238831 a(n) = 0 if n <= 2; thereafter a(n) = A238827(n) + A238830(n-2).
%H A238831 Vincenzo Librandi, <a href="/A238831/b238831.txt">Table of n, a(n) for n = 1..1000</a>
%H A238831 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 q(n).
%H A238831 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-1,-7,-1,6,6,1,-1).
%F A238831 G.f.: -x^6*(x-1)*(2*x+1)*(x^2+x+1) / ((x+1)^2*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1)). - _Colin Barker_, Mar 20 2014
%p A238831 g:=proc(n) option remember; local t1; t1:=[2,3,6,14,34,84,208,515];
%p A238831 if n <= 7 then t1[n] else
%p A238831 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 A238831 [seq(g(n),n=1..32)]; # A238823
%p A238831 d:=proc(n) option remember; global g; local t1; t1:=[0,1];
%p A238831 if n <= 2 then t1[n] else
%p A238831 g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
%p A238831 [seq(d(n),n=1..32)]; # A238824
%p A238831 p:=proc(n) option remember; global d; local t1; t1:=[0,0,0,1];
%p A238831 if n <= 4 then t1[n] else
%p A238831 p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;
%p A238831 [seq(p(n),n=1..32)]; # A238825
%p A238831 h:=n->p(n+3)-p(n+1); [seq(h(n),n=1..32)]; #A238826
%p A238831 r:=proc(n) option remember; global p; local t1; t1:=[0,0,0,0];
%p A238831 if n <= 4 then t1[n] else
%p A238831 r(n-2)+p(n-3); fi; end proc;
%p A238831 [seq(r(n),n=1..32)]; # A238827
%p A238831 b:=n-> if n=1 then 0 else d(n-1)+p(n); fi; [seq(b(n),n=1..32)]; #A238828
%p A238831 a:=n->g(n)-h(n); [seq(a(n),n=1..32)]; #A238829
%p A238831 i:=proc(n) option remember; global b,r; local t1; t1:=[0,0];
%p A238831 if n <= 2 then t1[n] else
%p A238831 i(n-2)+b(n-1)+r(n); fi; end proc;
%p A238831 [seq(i(n),n=1..32)]; # A238830
%p A238831 q:=n-> if n<=2 then 0 else r(n)+i(n-2); fi;
%p A238831 [seq(q(n),n=1..45)]; # A238831
%t A238831 CoefficientList[Series[- x^5 (x - 1) (2 x + 1) (x^2 + x + 1)/((x + 1)^2 (x^7 - 3 x^6 - x^5 - x^4 + 4 x^3 - 3 x + 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 21 2014 *)
%o A238831 (PARI) concat([0,0,0,0,0], Vec(-x^6*(x-1)*(2*x+1)*(x^2+x+1)/((x+1)^2*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1)) + O(x^100))) \\ _Colin Barker_, Mar 20 2014
%o A238831 (Magma) m:=40; R<x>:=LaurentSeriesRing(RationalField(), m); [0,0,0,0,0] cat Coefficients(R! -x^6*(x-1)*(2*x+1)*(x^2+x+1) / ((x+1)^2*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1))); // _Vincenzo Librandi_, Mar 21 2014
%Y A238831 Cf. A238823-A238830.
%K A238831 nonn,easy
%O A238831 1,7
%A A238831 _N. J. A. Sloane_, Mar 08 2014