cp's OEIS Frontend

A351323 Number of tilings of a 6 X n rectangle with right trominoes.

%I A351323 #24 Dec 08 2022 11:43:56
%S A351323 1,0,4,8,18,72,162,520,1514,4312,13242,39088,118586,361712,1103946,
%T A351323 3403624,10513130,32614696,101530170,316770752,990771834,3104283168,
%U A351323 9741133578,30606719000,96263812906,303028237848,954563802106,3008665176560,9487377712634,29928407213328
%N A351323 Number of tilings of a 6 X n rectangle with right trominoes.
%C A351323 See A351322 for algorithm. The subsequence 1,8,162,... for 6 X 3n rectangles also has a depending recurrence with 11 parameters.
%C A351323 The sequence is the Hadamard sum of the following 4 sequences: 0, 0, 0, 0, 16, 0, 128, 0, 256, 768, 1024,0, 13440, 0, 16384, .. (tilings which have both horizontal and vertical faults), 0, 0, 4, 8, 0, 0, 16, 0, 0, 128, 0, 0, 1536, 0, 0,.. (tilings which have horizontal faults but no vertical faults), 0, 0, 0, 0, 0, 64, 16, 480, 1140, 3200, 11208, 36032, 95924, 333856, 1003096,.. (tilings which have vertical faults but no horizontal faults), 1, 0, 0, 0, 2, 8, 2, 40, 118, 216, 1010, 3056, 7686, 27856, 84466,... (tilings which have neither vertical nor horizontal faults). - _R. J. Mathar_, Dec 08 2022
%H A351323 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (2,8,2,-43,-42,36,102,0,-44,-8,-8).
%F A351323 G.f.: (1 - x)*(1 - x - 5*x^2 - 7*x^3 + 6*x^4 + 12*x^5 + 6*x^6)/(1 - 2*x - 8*x^2 - 2*x^3 + 43*x^4 + 42*x^5 - 36*x^6 - 102*x^7 + 44*x^9 + 8*x^10 + 8*x^11).
%F A351323 a(n) = Sum_{i=0..10} b(i)*a(n-11+i) for n>10 where {b(i)} = {-8,-8,-44,0,102,36,-42,-43,2,8,2}.
%e A351323 For a 6 X 2 rectangle there are 4 tilings:
%e A351323    ___   ___   ___   ___
%e A351323   |  _| |  _| |_  | |_  |
%e A351323   |_| | |_| | | |_| | |_|
%e A351323   |___| |___| |___| |___|
%e A351323   |  _| |_  | |  _| |_  |
%e A351323   |_| | | |_| |_| | | |_|
%e A351323   |___| |___| |___| |___|
%Y A351323 Cf. A077957, A000079, A046984, A084478, A351322, A351324, A236576 (straight trominoes), A233290 (mixed trominoes).
%K A351323 nonn
%O A351323 0,3
%A A351323 _Gerhard Kirchner_, Feb 21 2022