A084478 Number of tilings of a 5 X 3n rectangle with right trominoes.

1, 0, 72, 384, 8544, 76800, 1168512, 12785664, 170678784, 2014648320, 25633231872, 311423852544, 3892030055424, 47803588208640, 593425578949632, 7318730222874624, 90624271197041664, 1119402280975349760, 13847850677651745792, 171150049715628539904

Offset: 0

Author

Ralf Stephan, May 27 2003

Keywords

• nonn
• easy

Comments

A right 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

There is a sign typo with respect to the g.f. in the paper.

The sequence is the Hadamard sum of the following 4 sequences: 0, 0, 0, 0, 2048, 0, 65536, 0,.. (tilings which have both vertical and horizontal faults), 0, 0, 64, 0, 0, 0, 0, 0.. (tilings which have horizontal but no vertical faults), 0, 0, 0, 0, 3136, 55296, 939008, 11649024... (tilings which have vertical faults but no horizontal faults), .. 1, 0, 8, 384, 3360, 21504, 163968 (essentially A084479) which have neither vertical nor horizontal faults. - R. J. Mathar, Dec 08 2022

Links

• Colin Barker, Table of n, a(n) for n = 0..900
• D. Merlini, R. Sprugnoli, M. C. Verri, Strip tiling and regular grammars, Theo. Comp. Sci. 242 (1-2) (2000) 109-124, Proof of Theorem 4.2 (typo t^5 in the denominator of g.f. ought be t^6)
• C. Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.
• Index entries for linear recurrences with constant coefficients, signature (2,103,280,380).

Crossrefs

Cf. A046984, A084477, A084479 (INVERT transform), A084480, A084481,A351323, A351324, A236576 (straight trominoes), A233340 (mixed trominoes).

Programs

Mathematica

LinearRecurrence[{2, 103, 280, 380}, {72, 384, 8544, 76800}, 20] (* Jean-François Alcover, Jan 07 2019 *)

PARI

Vec(24*x^2*(3 + 10*x + 15*x^2) / (1 - 2*x - 103*x^2 - 280*x^3 - 380*x^4) + O(x^30)) \\ Colin Barker, Mar 27 2017

Formula

G.f.: (1 - 2*z - 31*z^2 - 40*z^3 - 20*z^4) / (1 - 2*z - 103*z^2 - 280*z^3 - 380*z^4).

a(n) = 2*a(n-1) + 103*a(n-2) + 280*a(n-3) + 380*a(n-4) for n > 4. - Colin Barker, Mar 27 2017

Extensions

a(0) and a(1) prepended by Alois P. Heinz, Feb 21 2022