A332050 Number of ways to arrange Palago tiles in a triangle of side length n, up to rotation, reflection, and swapping colors.

1, 1, 7, 129, 9882, 2391930, 1743402771, 3812799008214, 25015772571200361, 492385451093553791610, 29074868501520453489499806, 5150525730438768829942800034449, 2737200544710109691113626131721984885, 4363981784043856212945753449232929426200329

Offset

0,3

Comments

A Palago tile is a hexagonal tile with four regions of alternating colors. See links for illustrations.

Links

Peter Kagey, Table of n, a(n) for n = 0..64

Code Golf Stack Exchange, Counting creatures on a hexagonal tiling

Peter Kagey, Example of for n = 2.

Formula

a(n) = (3^A000217(n) + 3*3^A002620(n) + 2*3^A007997(n+4))/6 if n = 1 (mod 3), and

a(n) = (3^A000217(n) + 3*3^A002620(n))/6 otherwise.

Mathematica

a[n_] = (3^Binomial[n + 1, 2] +
    3*3^((Binomial[n + 1, 2] - Ceiling[n/2])/2) +
    If[Mod[n, 3] == 1, 0, 2*3^(Binomial[n + 1, 2]/3)])/6

Crossrefs

Cf. A000217, A002620, A007997.

Cf. A325936.

Sequence in context: A095885 A361369 A361367 * A134056 A123036 A142011

Adjacent sequences: A332047 A332048 A332049 * A332051 A332052 A332053

Keyword

nonn

Author

Peter Kagey, Feb 06 2020

Status

approved