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A320494 Coordination sequence of thinnest 5-neighbor packing of the plane with congruent triangles with respect to a tetravalent point. 8
1, 4, 11, 15, 22, 28, 32, 39, 45, 48, 56, 62, 65, 73, 78, 82, 90, 95, 99, 106, 112, 116, 123, 129, 132, 140, 146, 149, 157, 162, 166, 174, 179, 183, 190, 196, 200, 207, 213, 216, 224, 230, 233, 241, 246, 250, 258, 263, 267, 274, 280, 284, 291, 297, 300, 308 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

"5-neighbor" means that each triangle has a point in common with exactly five other triangles.

This packing is actually the thinnest 5-neighbor packing in the plane using any congruent convex polygons.

More formally, this sequence is the coordination sequence of the vertex-edge graph of the packing with respect to a tetavalent vertex. The base vertex is marked "B" in the figure (it is the midpoint of an edge of the large empty triangle).

REFERENCES

William Moser and Janos Pach, Research Problems in Discrete Geometry: Packing and Covering, DIMACS Technical Report 93-32, May 1993. See Fig. 19.1a, page 32. There is an error in the figure: the triangle at the right of the bottom row should not be shaded. The figure shown here is correct.

LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..1000

Rémy Sigrist, Illustration of first terms

Rémy Sigrist, PARI program for A320494

N. J. A. Sloane, The packing and its graph. (The triangles are shaded, the base point is marked B, and the green dots indicate the centers of large empty triangles.)

FORMULA

Conjectures from Colin Barker, Oct 23 2018: (Start)

G.f.: (1 + 4*x + 11*x^2 + 14*x^3 + 18*x^4 + 16*x^5 + 13*x^6 + 6*x^7 + 3*x^8 - 2*x^9) / ((1 - x)^2*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)).

a(n) = a(n-3) + a(n-5) - a(n-8) for n>9.

(End)

PROG

(PARI) See Links section.

CROSSREFS

Cf. A320492, A320493, A320495, A320496, A320497, A320498.

Sequence in context: A032822 A288316 A003250 * A190509 A022131 A091391

Adjacent sequences:  A320491 A320492 A320493 * A320495 A320496 A320497

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 21 2018

EXTENSIONS

More terms from Rémy Sigrist, Oct 22 2018

STATUS

approved

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Last modified November 19 11:09 EST 2019. Contains 329319 sequences. (Running on oeis4.)