A084477 Number of fault-free tilings of a 4 X 3n rectangle with right trominoes.

4, 2, 8, 48, 288, 1728, 10368, 62208, 373248, 2239488, 13436928, 80621568, 483729408, 2902376448, 17414258688, 104485552128, 626913312768, 3761479876608, 22568879259648, 135413275557888, 812479653347328, 4874877920083968, 29249267520503808

Offset: 1

Author

Ralf Stephan, May 27 2003

Keywords

• nonn
• easy

Comments

A tromino is a 3-celled L-shaped piece (a 2 X 2 square with one of the four cells omitted). - N. J. A. Sloane, Mar 28 2017

Fault-free tilings are those where the only straight interface is at the left and right end. Thus a(n) <= A046984(n).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• M. Aanjaneya and S. P. Pal, Faultfree tromino tilings of rectangles
• C. Moore, [math/9905012] Some Polyomino Tilings of the Plane
• Index entries for linear recurrences with constant coefficients, signature (6).

Crossrefs

Cf. A084478, A084479, A084480, A084481.

Programs

PARI

Vec(2*x*(2 - 11*x - 2*x^2) / (1 - 6*x) + O(x^30)) \\ Colin Barker, Mar 28 2017

Formula

a(n) = 2*A067411(n-2) for n>1.

G.f.: 2*z(2-11*z-2*z^2) / (1-6*z).

a(n) = 8 * 6^(n-3) for n>2.

G.f.: 9/2 - x - 1/Q(0) where Q(k)= 1 + 5^k/(1 - 2*x/(2*x + 5^k/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 10 2013

a(n) = 6*a(n-1) for n>2. - Colin Barker, Mar 28 2017