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A385868 Number of ways to tile a hexagonal strip made up of n equilateral triangles, using triangles, diamonds, and trapezoids.

%I A385868 #20 Jul 22 2025 10:31:36
%S A385868 1,1,2,4,7,13,39,66,110,200,604,1032,1741,3149,9476,16202,27337,49461,
%T A385868 148841,254466,429308,776774,2337580,3996430,6742361,12199339,
%U A385868 36711974,62764458,105889743,191592331,576566591,985724436,1663012914,3008983882,9055057632,15480937786
%N A385868 Number of ways to tile a hexagonal strip made up of n equilateral triangles, using triangles, diamonds, and trapezoids.
%C A385868 Here is the hexagonal strip:
%C A385868     ________________      ____
%C A385868    /\  /\  /\  /\  /      \  /
%C A385868   /__\/__\/__\/__\/  ...   \/
%C A385868   \  /\  /\  /\  /\        /\
%C A385868    \/__\/__\/__\/__\      /__\
%C A385868 The three types of tiles are triangles, diamonds, and trapezoids (each of which can be rotated). Here are the three types of tiles:
%C A385868   ____        ____             ____
%C A385868   \  /        \   \           /    \
%C A385868    \/   and    \___\   and   /______\.
%C A385868 Compare to A356622 and A356623, which are on a similar board but only use triangles and diamonds.
%H A385868 <a href="/index/Rec#order_24">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,15,0,0,0,9,0,0,0,32,0,0,0,9,0,0,0,1,0,0,0,-1).
%F A385868 a(n) = 15*a(n-4) + 9*a(n-8) + 32*a(n-12) + 9*a(n-16) + 1*a(n-20)  - 1*a(n-24).
%F A385868 a(4*n+2) = t(2*n+1)^2 + t(2*n)^2 + 2*t(2*n)*t(2*n-1) + a(4*n) + 2*a(4*n-1) + Sum_{i=1..n-1} a(4*i)*(t(2*(n-i))^2 + 2*t(2*(n-i))*t(2*(n-i)-1)) + Sum_{i=1..n-1} a(4*i-1)*2*t(2*(n-i))^2, for t(n) = A000073(n+2).
%F A385868 G.f.: (x^18-x^17+x^16-x^14+4*x^12-6*x^11-x^10+4*x^9+4*x^8-6*x^7-9*x^6+2*x^5+8*x^4-4*x^3 -2*x^2-x-1) / (-x^24+x^20+9*x^16+32*x^12+9*x^8+15*x^4-1). - _Alois P. Heinz_, Jul 21 2025
%e A385868 Here is one of the a(13) = 3149 possible tilings for this strip of 13 triangular cells:
%e A385868    ____________
%e A385868   /   /\   \   \
%e A385868  /__ /__\   \ __\
%e A385868  \      /\  /\
%e A385868   \____/__\/__\.
%t A385868 a[0] = 1; a[1] = 1; a[2] = 2; a[3] = 4; a[4] = 7; a[5] = 13; a[6] = 39; a[7] = 66;
%t A385868 a[n_] := a[n] = Switch[Mod[n, 4],
%t A385868     0, a[n-1] + a[n-3] + 2 a[n-4] + 3 a[n-5] + 2 a[n-6] + a[n-7],
%t A385868     1, a[n-1] + a[n-2] + a[n-4] + a[n-5] + a[n-6],
%t A385868     2, a[n-1] + a[n-2] + 3 a[n-3] + a[n-4] + 3 a[n-5] + 2 a[n-6] + a[n-7],
%t A385868     3, a[n-1] + a[n-2] + a[n-3] + a[n-4] + a[n-5] + a[n-6]];
%t A385868 Table[a[n], {n, 0, 40}]
%Y A385868 Cf. A356623, A356623.
%K A385868 nonn,easy
%O A385868 0,3
%A A385868 _Greg Dresden_ and Sean Choi, Jul 10 2025