cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

18, 198, 1298, 5778, 19602, 54758, 132498, 287298, 571538, 1060902
Offset: 0

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Author

Craig Knecht, Jan 22 2025

Keywords

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

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).

A381555 Triangle read by rows T(n,k) is the number of diamond coverings for a specific number of diamonds covering an even length row of triangles.

Original entry on oeis.org

1, 4, 1, 8, 4, 1, 13, 16, 4, 1, 19, 41, 24, 4, 1, 26, 85, 85, 32, 4, 1, 34, 155, 231, 145, 40, 4, 1, 43, 259, 532, 489, 221, 48, 4, 1, 53, 406, 1092, 1365, 891, 313, 56, 4, 1, 64, 606, 2058, 3333, 2926, 1469, 421, 64, 4, 1, 76, 870, 3630, 7359, 8294, 5551, 2255, 545, 72, 4
Offset: 0

Views

Author

Craig Knecht, Feb 27 2025

Keywords

Comments

The total number of ways the diamond can cover a single row of length(n) triangles is the Fibonacci series. This total can be subdivided into categories based on the number of covering diamonds. The number of categories increase with the length of the row providing the structure of the triangle (see illustrations in the link below).
The above process provides a way to subdivide the individual Fibonacci numbers.
Comparing the diamond covering of a row of triangles shown here to the diamond corona of a hexagon A380346 or a diamond A380416 may be instructive.
A381552 provides additional graphics that help explain the diamond covering.

Examples

			Triangle begins:
  1, 4;
  1, 8, 4;
  1, 13, 16, 4;
  1, 19, 41, 24, 4;
  1, 26, 85, 85, 32, 4;
  1, 34, 155, 231, 145, 40, 4;

Links

Crossrefs

Cf. A063496, A081219, A102083, A323847, A380346, A380416, A381552.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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