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A238829 a(n) = A238823(n) - A238826(n).

%I A238829 #16 Sep 08 2022 08:46:07
%S A238829 1,1,2,5,12,31,77,192,474,1170,2881,7097,17477,43050,106043,261235,
%T A238829 643552,1585421,3905750,9621993,23704161,58396118,143860974,354406732,
%U A238829 873093707,2150897733,5298813853,13053818630,32158552201,79223751853,195170567014,480809724213
%N A238829 a(n) = A238823(n) - A238826(n).
%H A238829 Vincenzo Librandi, <a href="/A238829/b238829.txt">Table of n, a(n) for n = 1..1000</a>
%H A238829 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 a(n).
%H A238829 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-4,1,1,3,-1).
%F A238829 G.f.: -x*(x-1)*(2*x^5+x^4+x^3-2*x^2-x+1) / ( 1-3*x+4*x^3-x^4-x^5-3*x^6+x^7 ). - _R. J. Mathar_, Mar 20 2014
%p A238829 g:=proc(n) option remember; local t1;
%p A238829 t1:=[2,3,6,14,34,84,208,515];
%p A238829 if n <= 7 then t1[n] else
%p A238829 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 A238829 [seq(g(n),n=1..32)]; # A238823
%p A238829 d:=proc(n) option remember; global g; local t1;
%p A238829 t1:=[0,1];
%p A238829 if n <= 2 then t1[n] else
%p A238829 g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
%p A238829 [seq(d(n),n=1..32)]; # A238824
%p A238829 p:=proc(n) option remember; global d; local t1;
%p A238829 t1:=[0,0,0,1];
%p A238829 if n <= 4 then t1[n] else
%p A238829 p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;
%p A238829 [seq(p(n),n=1..32)]; # A238825
%p A238829 h:=n->p(n+3)-p(n+1);
%p A238829 [seq(h(n),n=1..32)]; #A238826
%p A238829 r:=proc(n) option remember; global p; local t1;
%p A238829 t1:=[0,0,0,0];
%p A238829 if n <= 4 then t1[n] else
%p A238829 r(n-2)+p(n-3); fi; end proc;
%p A238829 [seq(r(n),n=1..32)]; # A238827
%p A238829 [0,seq(d(n-1)+p(n),n=2..32)]; #A238828
%p A238829 a:=n->g(n)-h(n);
%p A238829 [seq(a(n),n=1..32)]; #A238829
%t A238829 CoefficientList[Series[(1 - x) (2 x^5 + x^4 + x^3 - 2 x^2 - x + 1)/(1 - 3 x + 4 x^3 - x^4 - x^5 - 3 x^6 + x^7), {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 21 2014 *)
%t A238829 LinearRecurrence[{3,0,-4,1,1,3,-1},{1,1,2,5,12,31,77},40] (* _Harvey P. Dale_, Jun 08 2018 *)
%o A238829 (Magma) I:=[1,1,2,5,12,31,77]; [n le 7 select I[n] else 3*Self(n-1)-4*Self(n-3)+Self(n-4)+Self(n-5)+3*Self(n-6)-Self(n-7): n in [1..35]]
%Y A238829 Cf. A238823-A238828.
%K A238829 nonn,easy
%O A238829 1,3
%A A238829 _N. J. A. Sloane_, Mar 08 2014