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A238826 a(n) = p(n+3)-p(n+1), where p(n) = A238825(n).

%I A238826 #20 Aug 15 2025 12:43:43
%S A238826 1,2,4,9,22,53,131,323,798,1968,4853,11958,29463,72581,178803,440474,
%T A238826 1085110,2673183,6585468,16223521,39967243,98460769,242561730,
%U A238826 597559646,1472109847,3626595728,8934249307,22009844973,54222045921,133577963318,329074124992,810685962909
%N A238826 a(n) = p(n+3)-p(n+1), where p(n) = A238825(n).
%H A238826 Vincenzo Librandi, <a href="/A238826/b238826.txt">Table of n, a(n) for n = 1..1000</a>
%H A238826 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 h(n).
%H A238826 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-4,1,1,3,-1).
%F A238826 G.f.: -x*(1+x)*(x^3+x^2-1)*(x-1)^2 / ( 1-3*x+4*x^3-x^4-x^5-3*x^6+x^7 ). - _R. J. Mathar_, Mar 20 2014
%p A238826 g:=proc(n) option remember; local t1;
%p A238826 t1:=[2,3,6,14,34,84,208,515];
%p A238826 if n <= 7 then t1[n] else
%p A238826 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 A238826 [seq(g(n),n=1..32)]; # A238823
%p A238826 d:=proc(n) option remember; global g; local t1;
%p A238826 t1:=[0,1];
%p A238826 if n <= 2 then t1[n] else
%p A238826 g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
%p A238826 [seq(d(n),n=1..32)]; # A238824
%p A238826 p:=proc(n) option remember; global d; local t1;
%p A238826 t1:=[0,0,0,1];
%p A238826 if n <= 4 then t1[n] else
%p A238826 p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;
%p A238826 [seq(p(n),n=1..32)]; # A238825
%p A238826 [seq(p(n+3)-p(n+1),n=1..32)]; #A238826
%t A238826 CoefficientList[Series[-(1 + x) (x^3 + x^2 - 1) (x - 1)^2/(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 A238826 LinearRecurrence[{3,0,-4,1,1,3,-1},{1,2,4,9,22,53,131},40] (* _Harvey P. Dale_, Aug 15 2025 *)
%o A238826 (Magma) m:=40; R<x>:=LaurentSeriesRing(RationalField(), m); Coefficients(R! -x*(1+x)*(x^3+x^2-1)*(x-1)^2 / ( 1-3*x+4*x^3-x^4-x^5-3*x^6+x^7)); // _Vincenzo Librandi_, Mar 21 2014
%Y A238826 Cf. A238823-A238825.
%K A238826 nonn,easy
%O A238826 1,2
%A A238826 _N. J. A. Sloane_, Mar 08 2014