A298040 Coordination sequence of Dual(4.6.12) tiling with respect to a tetravalent node.

1, 4, 20, 24, 40, 40, 60, 56, 80, 72, 100, 88, 120, 104, 140, 120, 160, 136, 180, 152, 200, 168, 220, 184, 240, 200, 260, 216, 280, 232, 300, 248, 320, 264, 340, 280, 360, 296, 380, 312, 400, 328, 420, 344, 440, 360, 460, 376, 480, 392, 500, 408, 520, 424, 540

Offset

0,2

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Tom Karzes, Tiling Coordination Sequences

N. J. A. Sloane, Illustration of initial terms (shows one 90-degree quadrant of tiling)

N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Formula

Conjecture: For n>0, a(n)=10n if n even, otherwise 8n.

Conjectures from Colin Barker, Apr 01 2020: (Start)

G.f.: (1 + 4*x + 18*x^2 + 16*x^3 + x^4 - 4*x^5) / ((1 - x)^2*(1 + x)^2).

a(n) = (9 + (-1)^n)*n for n>1.

a(n) = 2*a(n-2) - a(n-4) for n>5.

(End)

Mathematica

LinearRecurrence[{0, 2, 0, -1}, {1, 4, 20, 24, 40, 40}, 60] (* Harvey P. Dale, Apr 06 2022 *)

Crossrefs

Cf. A072154, A298041 (partial sums), A298036 (12-valent node), A298038 (hexavalent node).

List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

Sequence in context: A273791 A065984 A039569 * A174134 A032425 A194045

Adjacent sequences: A298037 A298038 A298039 * A298041 A298042 A298043

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 22 2018

Extensions

Terms a(8)-a(54) added by Tom Karzes, Apr 01 2020

Status

approved