A354541 Number of ways to tile a double-hexagon strip of n hexagons, using single and double hexagons.

1, 1, 2, 4, 8, 12, 24, 48, 72, 144, 288, 432, 864, 1728, 2592, 5184, 10368, 15552, 31104, 62208, 93312, 186624, 373248, 559872, 1119744, 2239488, 3359232, 6718464, 13436928, 20155392, 40310784, 80621568, 120932352, 241864704, 483729408

Offset

0,3

Comments

Here is the hexagonal strip for n=9:

____ ____ ____

/ \ / \ / \

____/ 2 \____/ 5 \____/ 8 \

/ \ / \ / \ /

/ 1 \____/ 4 \____/ 7 \____/

\ / \ / \ / \

\____/ 3 \____/ 6 \____/ 9 \

\ / \ / \ /

\____/ \____/ \____/

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,6).

Formula

a(3*n) = a(3*n-1) + a(3*n-2) + a(3*n-3) = 4*6^(n-1).

a(3*n+1) = a(3*n) + 2*a(3*n-1) = 8*6^(n-1).

a(3*n+2) = a(3*n+1) + a(3*n) = 12*6^(n-1).

From Stefano Spezia, Aug 28 2022: (Start)

G.f.: (1 + x + 2*x^2 - 2*x^3 +2*x^4)/(1 - 6*x^3).

a(n) = 6*a(n-3) for n > 4. (End)

From Amiram Eldar, May 28 2024: (Start)

Sum_{n>=0} 1/a(n) = 61/20.

Sum_{n>=0} (-1)^n/a(n) = 9/28. (End)

Example

Here is one of the 144 tilings for n=9, this one using three singles and three doubles:
          ____        ____        ____
         /    \      /    \      /    \
    ____/      \____/      \____/      \
   /           /    \      /    \      /
  /       ____/      \    /      \____/
  \      /    \      /    \           \
   \____/      \____/      \____       \
        \      /    \      /    \      /
         \____/      \____/      \____/

Mathematica

LinearRecurrence[{0, 0, 6}, {1, 1, 2, 4, 8}, 50] (* Paolo Xausa, May 27 2024 *)

Crossrefs

Sequence in context: A181808 A343014 A097942 * A358513 A004653 A115386

Adjacent sequences: A354538 A354539 A354540 * A354542 A354543 A354544

Keyword

nonn,easy

Author

Greg Dresden and Zeno Changze Song, Aug 17 2022

Status

approved