%I #21 Feb 09 2024 17:20:47
%S 0,1,0,2,2,7,16,40,101,246,615,1504,3724,9147,22567,55541,136884,
%T 337128,830628,2046145,5040932,12418320,30593281,75367352,185670647,
%U 457405836,1126836394,2776001211,6838779857,16847579205,41504619640,102248123906,251891939366,620544865783,1528734638988,3766092860744
%N a(n) = n-1 for n <= 2; thereafter a(n) = A238824(n-2) + A238832(n-1).
%H Vincenzo Librandi, <a href="/A238833/b238833.txt">Table of n, a(n) for n = 1..1000</a>
%H 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 j(n).
%H <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 G.f.: -x^2*(x^8+2*x^7+x^6-2*x^5-2*x^4-x^3+3*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 g:=proc(n) option remember; local t1; t1:=[2,3,6,14,34,84,208,515];
%p if n <= 7 then t1[n] else
%p 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 [seq(g(n),n=1..32)]; # A238823
%p d:=proc(n) option remember; global g; local t1; t1:=[0,1];
%p if n <= 2 then t1[n] else
%p g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
%p [seq(d(n),n=1..32)]; # A238824
%p p:=proc(n) option remember; global d; local t1; t1:=[0,0,0,1];
%p if n <= 4 then t1[n] else
%p p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;
%p [seq(p(n),n=1..32)]; # A238825
%p h:=n->p(n+3)-p(n+1); [seq(h(n),n=1..32)]; #A238826
%p r:=proc(n) option remember; global p; local t1; t1:=[0,0,0,0];
%p if n <= 4 then t1[n] else
%p r(n-2)+p(n-3); fi; end proc;
%p [seq(r(n),n=1..32)]; # A238827
%p b:=n-> if n=1 then 0 else d(n-1)+p(n); fi; [seq(b(n),n=1..32)]; #A238828
%p a:=n->g(n)-h(n); [seq(a(n),n=1..32)]; #A238829
%p i:=proc(n) option remember; global b,r; local t1; t1:=[0,0];
%p if n <= 2 then t1[n] else
%p i(n-2)+b(n-1)+r(n); fi; end proc;
%p [seq(i(n),n=1..32)]; # A238830
%p q:=n-> if n<=2 then 0 else r(n)+i(n-2); fi;
%p [seq(q(n),n=1..45)]; # A238831
%p e:=n-> if n<=1 then 0 else d(n-1)+i(n-1); fi;
%p [seq(e(n),n=1..45)]; # A238832
%p j:=n-> if n<=2 then n-1 else d(n-2)+e(n-1); fi;
%p [seq(j(n),n=1..45)]; # A238833
%t CoefficientList[Series[- x (x^8 + 2 x^7 + x^6 - 2 x^5 - 2 x^4 - x^3 + 3 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 *)
%t LinearRecurrence[{1,5,-1,-7,-1,6,6,1,-1},{0,1,0,2,2,7,16,40,101,246},40] (* _Harvey P. Dale_, Jul 23 2021 *)
%o (PARI) concat(0, Vec(-x^2*(x^8+2*x^7+x^6-2*x^5-2*x^4-x^3+3*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 (Magma) m:=40; R<x>:=LaurentSeriesRing(RationalField(), m); [0] cat Coefficients(R! -x^2*(x^8+2*x^7+x^6-2*x^5-2*x^4-x^3+3*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 Cf. A238823-A238832.
%K nonn,easy
%O 1,4
%A _N. J. A. Sloane_, Mar 08 2014
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