Craig Knecht - cp's OEIS frontend

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.

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.

A381552 Triangle read by rows T(n,k) is the number of diamond coverings for a specific number of diamonds covering an odd length row of triangles.

Original entry on oeis.org

3, 4, 4, 5, 12, 4, 6, 25, 20, 4, 7, 44, 61, 28, 4, 8, 70, 146, 113, 36, 4, 9, 104, 301, 344, 181, 44, 4, 10, 147, 560, 876, 670, 265, 52, 4, 11, 200, 966, 1968, 2035, 1156, 365, 60, 4, 12, 264, 1572, 4026, 5368, 4082, 1834, 481, 68, 4, 13, 340, 2442, 7656, 12727, 12376, 7385, 2736, 613, 76, 4, 14, 429, 3652, 13728, 27742, 33397, 25312, 12376, 3894, 761, 84, 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).

A381555 provides additional graphics explaining the diamond coverings.

			Triangle begins:
  3;
  4,  4;
  5, 12,   4;
  6, 25,  20,   4;
  7, 44,  61,  28,  4;
  8, 70, 146, 113, 36, 4;
  ...

Craig Knecht, Splitting the Fibonacci numbers.
Craig Knecht, Fibonacci Expansion.

Cf. A000297, A080856, A381555.

A380608 a(n) is the number of distinct ways to cut a hexagon with edges of size n into diamonds with integer sides.

Original entry on oeis.org

2, 37, 6330, 12807773

Offset: 0

Craig Knecht, Jan 28 2025

The number of ways to cut a diamond with edges of size n into diamonds with integer sides is A045846.

Craig Knecht, Diamond division of the hexagon.

Cf. A045846.

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

Original entry on oeis.org

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.

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

Craig Knecht, Jan 22 2025

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.

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.

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

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

A378455 Total number of coronal tilings of a width one length n straight polyiamond central frame with a specific hexiamond tile.

Original entry on oeis.org

1272, 2644, 2684, 3141, 3144, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185, 3184, 3185

Offset: 1

Craig Knecht, Nov 26 2024

For even length n>4 the total number of coronal tilings is 3185.
For odd length n>5 the total number of coronal tilings is 3184.
The corona can be composed of different numbers of coronal tiles. The number of coronal tilings for a given number of coronal tiles is noted.

Craig Knecht, Coronal detail.
Craig Knecht, Example for the sequence.
Craig Knecht, n=9, 16 coronal tiles, 72 tilings.
Craig Knecht, Repeating 3185 3184 coronal tiling totals.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Drop[CoefficientList[Series[x*(1272 + 2644*x + 1412*x^2 + 497*x^3 + 460*x^4 + 44*x^5 + 40*x^6)/(1 - x^2), {x,0,32}], x], 1] (* Georg Fischer, Feb 11 2025 *)

G.f.: x*(1272 + 2644*x + 1412*x^2 + 497*x^3 + 460*x^4 + 44*x^5 + 40*x^6)/(1 - x^2). - Stefano Spezia, Nov 27 2024

A377978 Total number of coronal tilings for a width one length n straight polyiamond with the T2 triangle.

Original entry on oeis.org

2, 2, 6, 11, 6, 2, 6, 11, 6, 2, 6, 11, 6, 2

Offset: 1

Craig Knecht, Nov 26 2024

Heesch numbers look at the buildout of coronal layers from a single type of tile. In contrast this sequence looks at a single type of tile being asked to form a corona around an expanding central frame.
This central frame tiled with the hexiamond boat tile has a coronal expansion number of 12 (see links below).
A376899 shows coronal tilings for the expansion of a central triangle frame and gives the definition for the coronal expansion number.
Combining individual polyiamonds in various ways creates a variety of expansion frames.Craig Knecht, Nov 29 2024

Craig Knecht, example for the sequence.
Craig Knecht, Expansion frame worksheet.
Craig Knecht, Hexiamond boat corona.

Cf. A376899.

A376899 Total number of ways a triangle of order n can be completely surrounded by a specific pentiamond.

Original entry on oeis.org

160, 24, 27, 8, 0, 0, 0

Offset: 1

Craig Knecht, Oct 08 2024

The coronal expansion number is defined to be the number of expansions the central host frame can undergo and still have a corona formed by the coronal tile. The coronal expansion number is 4 for this sequence. See the links section for an example of a pseudo-triangle as the central host frame that has a coronal expansion number of 7. Craig Knecht, Oct 31 2024

Craig Knecht, Pseudo triangle with a coronal expansion number of 7.
Craig Knecht, The 160 ways this pentiamond can surround a T1 triangle.
Craig Knecht, T2, T3, and T4 surrounded by this pentiamond.

A367703 a(n) = the maximum number of distinct compositions of a polycube of size n using any combination of polycubes of sizes 1 through n.

Original entry on oeis.org

1, 2, 4, 12, 24, 74, 228, 958, 1916, 6020, 18870

Offset: 1

John Mason from an idea of Craig Knecht, Nov 27 2023

The sequence considers reflections and rotations as distinct compositions.

Cf. A367173.

A367174 a(n) = the maximum number of distinct tilings of a polyhex of size n using any combination of polyhex tiles of sizes 1 through n.

Original entry on oeis.org

1, 2, 5, 13, 26, 89, 233, 610, 2179, 5707

Offset: 1

John Mason from an idea of Craig Knecht, Nov 07 2023

The sequence considers reflections and rotations as distinct tilings. The polyhexes being tiled and the tiles themselves may be with or without holes.

			a(3) = 5 because the polyhex of size 3 that has each cell touching both the other cells can be tiled by (1) a polyhex of size 3, (2) 3 polyhexes of size 1, and (3) a polyhex of size 1 and a polyhex of size 2 in three distinct ways.

Cf. A000228, A367172, A367173.

cp's OEIS Frontend

User: Craig Knecht

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.

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A381552 Triangle read by rows T(n,k) is the number of diamond coverings for a specific number of diamonds covering an odd length row of triangles.

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A380608 a(n) is the number of distinct ways to cut a hexagon with edges of size n into diamonds with integer sides.

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

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

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A378455 Total number of coronal tilings of a width one length n straight polyiamond central frame with a specific hexiamond tile.

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A377978 Total number of coronal tilings for a width one length n straight polyiamond with the T2 triangle.

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A376899 Total number of ways a triangle of order n can be completely surrounded by a specific pentiamond.

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A367703 a(n) = the maximum number of distinct compositions of a polycube of size n using any combination of polycubes of sizes 1 through n.

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A367174 a(n) = the maximum number of distinct tilings of a polyhex of size n using any combination of polyhex tiles of sizes 1 through n.

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