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

0,1

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 \___\.

Links

Table of n, a(n) for n=0..18.

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

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

Example

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

Mathematica

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

Crossrefs

Cf. A356622, A190984.

Sequence in context: A037565 A329170 A125835 * A091170 A207319 A361961

Adjacent sequences: A356620 A356621 A356622 * A356624 A356625 A356626

Keyword

nonn

Author

Greg Dresden and Aarnav Gogri, Aug 17 2022

Status

approved