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Number of ways to tile a hexagonal strip made up of n equilateral triangles, using singles and triples.
1

%I #11 Sep 17 2023 02:00:55

%S 1,1,1,2,3,4,10,16,23,29,70,113,166,211,505,813,1191,1515,3631,5846,

%T 8566,10894,26105,42032,61590,78330,187700,302216,442837,563199,

%U 1349583,2172966,3184049,4049464,9703648,15623862,22893659,29116089,69770292,112337277

%N Number of ways to tile a hexagonal strip made up of n equilateral triangles, using singles and triples.

%C Here is the hexagonal strip:

%C ________________ ____

%C /\ /\ /\ /\ / \ /

%C /__\/__\/__\/__\/ ... \/

%C \ /\ /\ /\ /\ /\

%C \/__\/__\/__\/__\ /__\

%C The two types of tiles are singles and triples (each of which can be rotated). Here are the two types of tiles:

%C ____ ________

%C \ / \ /\ /

%C \/ and \/__\/.

%H <a href="/index/Rec#order_24">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,7,0,0,0,1,0,0,0,3,0,0,0,-3,0,0,0,3,0,0,0,-1).

%F a(n) = 7*a(n-4) + a(n-8) + 3*a(n-12) - 3*a(n-16) + 3*a(n-20) - a(n-24).

%F a(4*n+2) = b(2*n+1)^2 + 2*Sum_{k=0..n} b(2*(n-k))*(a(4*k)*b(2*(n-k)-1) + a(4*k-1)*b(2*(n-k))), where b(n)=A000930(n), the Narayana's Cows sequence, so long as we agree that a(-1)=0 and b(-1)=0. - _Greg Dresden_, Sep 16 2023

%e For n=16, here is one of the a(16)=1191 ways to tile this strip (of 16 triangles) using singles and triples.

%e ________________

%e / / /\ /

%e / / /__\ __ /

%e \ /\ /\ /\

%e \/__\/__\___ /__\

%t g[-3] = 0;g[-2] = 0;g[-1] = 0;g[0] = 1;g[1] = 1;g[2] = 1;g[3] = 2;g[4] = 3;

%t g[n_] := g[n] = Switch[Mod[n, 4],

%t 0, g[n - 1] + g[n - 4] + g[n - 5] + g[n - 6] + g[n - 7],

%t 1, g[n - 1] + g[n - 4] + g[n - 6],

%t 2, g[n - 1] + 2 g[n - 3] + g[n - 5] + g[n - 6] + g[n - 7],

%t 3, g[n - 1] + g[n - 3] + g[n - 4] + g[n - 6]];

%t Table[g[n], {n, 0, 30}]

%Y Cf. A356622, A356623.

%K nonn

%O 0,4

%A _Greg Dresden_ and Yiwei Wang, Sep 02 2023