A356622 Number of ways to tile a hexagonal strip made up of 4*n equilateral triangles, using triangles and diamonds.

1, 5, 39, 317, 2585, 21085, 171987, 1402873, 11443033, 93339173, 761354199, 6210256613, 50656169297, 413195081581, 3370372805763, 27491645850097, 224245398092113, 1829137434684101, 14920010771362215

Offset

0,2

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) = A355327(2*n).

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

G.f.: (1 - 4 x + x^2)/(1 - 9 x + 7 x^2 - x^3).

Example

For n=4, here is one of the a(4)=2585 ways to tile this strip (of 16 triangles) using triangles and diamonds.
    ________________
   /   /\  /\  /   /
  /__ /  \/__\/__ /
  \  /\  /\   \  /\
   \/__\/__\___\/__\

Mathematica

LinearRecurrence[{9, -7, 1}, {1, 5, 39}, 40]

Crossrefs

Bisection of A355327. Cf. A356623.

Sequence in context: A201442 A135849 A105426 * A273019 A244039 A328554

Adjacent sequences: A356619 A356620 A356621 * A356623 A356624 A356625

Keyword

nonn,easy

Author

Greg Dresden and Aarnav Gogri, Aug 16 2022

Status

approved