A380346 -id:A380346 - cp's OEIS frontend

A380416 Number of corona for a diamond of edge n with diamonds of side 1.

18, 83, 258, 627, 1298, 2403, 4098, 6563, 10002

Offset: 0

Craig Knecht, Jan 24 2025

The number of diamonds that can surround a diamond(n) are the natural numbers > 2. These numbers fall into four categories: A004767(n), A004767(n) + 1, A004767(n) + 2, and A004767(n) + 3.
The number of coronal tilings for A004767(n) is 2.
The number of coronal tilings for A004767(n) + 1 is 9,25,49,81,121,169, see A016754.
The number of coronal tilings for A004767(n) + 2 is 6,40,126,288,550,936, see A089207.
The number of coronal tilings for A004767(n) + 3 is 1,16,81,256,625,1296, see A000583.

Craig Knecht, 83 corona of the D1 diamond.
Craig Knecht, Coronal Reptile.
Craig Knecht, Example for the sequence.
Craig Knecht, Feihu Liu, and Guoce Xin, Enumeration of Corona for Lozenge Tilings, arXiv:2504.19269 [math.CO], 2025.

Cf. A000583, A004767, A016754, A089207, A380346.

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

Craig Knecht, Feb 27 2025

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.

			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;

Craig Knecht, Diamond covering of a even length row of triangles.
Craig Knecht, Geometric bisection of the Fibonacci sequence.
Craig Knecht, Subdividing the individual Fibonacci numbers.
Walter Trump, Diamond covering producing the Fibonacci sequence.

Cf. A063496, A081219, A102083, A323847, A380346, A380416, A381552.

cp's OEIS Frontend

A380416 Number of corona for a diamond of edge n with diamonds of side 1.

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