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A084478
Number of tilings of a 5 X 3n rectangle with right trominoes.
9
1, 0, 72, 384, 8544, 76800, 1168512, 12785664, 170678784, 2014648320, 25633231872, 311423852544, 3892030055424, 47803588208640, 593425578949632, 7318730222874624, 90624271197041664, 1119402280975349760, 13847850677651745792, 171150049715628539904
OFFSET
0,3
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
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.
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
MATHEMATICA
LinearRecurrence[{2, 103, 280, 380}, {72, 384, 8544, 76800}, 20] (* Jean-François Alcover, Jan 07 2019 *)
PROG
(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
CROSSREFS
Cf. A046984, A084477, A084479 (INVERT transform), A084480, A084481,A351323, A351324, A236576 (straight trominoes), A233340 (mixed trominoes).
Sequence in context: A192792 A303621 A084479 * A187158 A164377 A156632
KEYWORD
nonn,easy
AUTHOR
Ralf Stephan, May 27 2003
EXTENSIONS
a(0) and a(1) prepended by Alois P. Heinz, Feb 21 2022
STATUS
approved