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A238832 a(1)=0; thereafter a(n) = A238824(n-1)+A238830(n-1).

%I A238832 #12 Feb 09 2024 17:20:59
%S A238832 0,0,1,1,4,9,23,58,141,353,861,2134,5236,12924,31798,78382,193029,
%T A238832 475619,1171600,2886427,7110657,17517598,43154977,106314193,261908415,
%U A238832 645221312,1589525242,3915853416,9646844896,23765351096,58546797181,144232146189,355321086856,875346302897,2156447153427,5312485264678
%N A238832 a(1)=0; thereafter a(n) = A238824(n-1)+A238830(n-1).
%H A238832 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 e(n).
%H A238832 <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 A238832 G.f.: x^3*(2*x^5+2*x^4+x^3-2*x^2+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 A238832 g:=proc(n) option remember; local t1; t1:=[2,3,6,14,34,84,208,515];
%p A238832 if n <= 7 then t1[n] else
%p A238832 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 A238832 [seq(g(n),n=1..32)]; # A238823
%p A238832 d:=proc(n) option remember; global g; local t1; t1:=[0,1];
%p A238832 if n <= 2 then t1[n] else
%p A238832 g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
%p A238832 [seq(d(n),n=1..32)]; # A238824
%p A238832 p:=proc(n) option remember; global d; local t1; t1:=[0,0,0,1];
%p A238832 if n <= 4 then t1[n] else
%p A238832 p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;
%p A238832 [seq(p(n),n=1..32)]; # A238825
%p A238832 h:=n->p(n+3)-p(n+1); [seq(h(n),n=1..32)]; #A238826
%p A238832 r:=proc(n) option remember; global p; local t1; t1:=[0,0,0,0];
%p A238832 if n <= 4 then t1[n] else
%p A238832 r(n-2)+p(n-3); fi; end proc;
%p A238832 [seq(r(n),n=1..32)]; # A238827
%p A238832 b:=n-> if n=1 then 0 else d(n-1)+p(n); fi; [seq(b(n),n=1..32)]; #A238828
%p A238832 a:=n->g(n)-h(n); [seq(a(n),n=1..32)]; #A238829
%p A238832 i:=proc(n) option remember; global b,r; local t1; t1:=[0,0];
%p A238832 if n <= 2 then t1[n] else
%p A238832 i(n-2)+b(n-1)+r(n); fi; end proc;
%p A238832 [seq(i(n),n=1..32)]; # A238830
%p A238832 q:=n-> if n<=2 then 0 else r(n)+i(n-2); fi;
%p A238832 [seq(q(n),n=1..45)]; # A238831
%p A238832 e:=n-> if n<=1 then 0 else d(n-1)+i(n-1); fi;
%p A238832 [seq(e(n),n=1..45)]; # A238832
%o A238832 (PARI) concat([0,0], Vec(x^3*(2*x^5+2*x^4+x^3-2*x^2+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
%Y A238832 Cf. A238823-A238831.
%K A238832 nonn,easy
%O A238832 1,5
%A A238832 _N. J. A. Sloane_, Mar 08 2014