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

2, 18, 148, 1208, 9854, 80378, 655632, 5347896, 43622018, 355818522, 2902360468, 23674136576, 193106524430, 1575142124306, 12848207584320, 104800979913168, 854846508252578, 6972859922465346, 56876614724333236

Offset: 0

Author

Greg Dresden and Aarnav Gogri, Aug 17 2022

Keywords

• nonn

Comments

Here is the hexagonal strip:

______________ __

/\ /\ /\ /\ / \ /\

/\/\/\/\/ ... \/\

\ /\ /\ /\ /\ /\ /

\/\/\/\/\ /\/

The two types of tiles are triangles and diamonds (each of which can be rotated). Here are the two types of tiles:

__ __

\ / \ \

\/ and \_\.

Examples

			For n=3, here is one of the a(3)=1208 ways to tile this strip (of 14 triangles) using triangles and diamonds.
    ____________
   /\  /\   \   \
  /__\/  \___\ __\
  \  /\  /   /\  /
   \/__\/__ /__\/
		

Links

• Index entries for linear recurrences with constant coefficients, signature (9, -7, 1).

Crossrefs

Cf. A356622, A190984.

Programs

Mathematica

LinearRecurrence[{9, -7, 1}, {2, 18, 148}, 40]

Formula

a(n) = 9*a(n-1) - 7*a(n-2) + a(n-3).

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).

G.f.: 2/(1 - 9*x + 7*x^2 - x^3).

a(n) = 2 + a(n-1) + 2*Sum_{k=1..n}(a(k-1)+A356622(k)). - Aarnav Gogri, Aug 17 2022

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