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

Author

N. J. A. Sloane, Jan 22 2018

Keywords

• nonn
• easy

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).

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.

Programs

Mathematica

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

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)

Extensions

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