A380346 Number of corona for a hexagon of edge n with diamonds of side 1.

18, 198, 1298, 5778, 19602, 54758, 132498, 287298, 571538, 1060902

Offset: 0

Author

Craig Knecht, Jan 22 2025

Keywords

• nonn
• more

Comments

The number of diamonds that can surround a hexagon(n) fall into four categories: A016945(n), A016945(n) + 1, A016945(n) + 2, and A016945(n) + 3.

The number of coronal tilings for A016945(n) is 2.

The number of coronal tilings for A016945(n) + 1 is 9,36,81,144,225, see A016766.

The number of coronal tilings for A016945(n) + 2 is 6,96,486,1536,3750,7776,14406 = 6*A000583.

The number of coronal tilings for A016945(n) + 3 is 1,64,729,4096,15625, see A001014.

A008793 looks at the enumeration of diamonds inside the hexagon. In contrast this looks at the enumeration of diamond corona of the hexagon.

Links

• Craig Knecht, 198 diamond corona of the H1 hexagon.
• Craig Knecht, Coronal corners.
• Craig Knecht, Example for the sequence.
• Craig Knecht, H1 corona sorted by the number of coronal tiles.
• Craig Knecht, Hexagon diamond corona inside a hexagon diamond tiling.
• Craig Knecht, Feihu Liu, and Guoce Xin, Enumeration of Corona for Lozenge Tilings, arXiv:2504.19269 [math.CO], 2025.
• Walter Trump, Lozenge-Coronae-Around-Hexagon.

Crossrefs

Cf. A001014, A008793, A016766, A016945, A380416.

Formula

a(n) = n^6 + 6*n^5 + 21*n^4 + 44*n^3 + 60*n^2 + 48*n + 18 (conjectured).