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A356623 Number of ways to tile a hexagonal strip made up of 4*n+2 equilateral triangles, using triangles and diamonds.

%I A356623 #26 Jul 04 2023 14:24:33
%S A356623 2,18,148,1208,9854,80378,655632,5347896,43622018,355818522,
%T A356623 2902360468,23674136576,193106524430,1575142124306,12848207584320,
%U A356623 104800979913168,854846508252578,6972859922465346,56876614724333236
%N A356623 Number of ways to tile a hexagonal strip made up of 4*n+2 equilateral triangles, using triangles and diamonds.
%C A356623 Here is the hexagonal strip:
%C A356623     ________________      ____
%C A356623    /\  /\  /\  /\  /      \  /\
%C A356623   /__\/__\/__\/__\/  ...   \/__\
%C A356623   \  /\  /\  /\  /\        /\  /
%C A356623    \/__\/__\/__\/__\      /__\/
%C A356623 The two types of tiles are triangles and diamonds (each of which can be rotated). Here are the two types of tiles:
%C A356623    ____        ____
%C A356623    \  /        \   \
%C A356623     \/   and    \___\.
%H A356623 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9, -7, 1).
%F A356623 a(n) = 9*a(n-1) - 7*a(n-2) + a(n-3).
%F A356623 a(n) = 2^(n+1) + Sum_{k=1..n} 2^(n-k)*(3*b(k) - b(k-1)) for n>=1, for b(n) = A356622(n).
%F A356623 G.f.: 2/(1 - 9*x + 7*x^2 - x^3).
%F A356623 a(n) = 2 + a(n-1) + 2*Sum_{k=1..n}(a(k-1)+A356622(k)). - _Aarnav Gogri_, Aug 17 2022
%F A356623 a(n+3) = 2*b(n+3) + Sum_{k=0..n} a(k)*b(n-k) for b(n) = A190984(n+1). - _Greg Dresden_ and _Aarnav Gogri_, Aug 24 2022
%e A356623 For n=3, here is one of the a(3)=1208 ways to tile this strip (of 14 triangles) using triangles and diamonds.
%e A356623     ____________
%e A356623    /\  /\   \   \
%e A356623   /__\/  \___\ __\
%e A356623   \  /\  /   /\  /
%e A356623    \/__\/__ /__\/
%t A356623 LinearRecurrence[{9, -7, 1}, {2, 18, 148}, 40]
%Y A356623 Cf. A356622, A190984.
%K A356623 nonn
%O A356623 0,1
%A A356623 _Greg Dresden_ and _Aarnav Gogri_, Aug 17 2022