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A351323
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Number of tilings of a 6 X n rectangle with right trominoes.
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4
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1, 0, 4, 8, 18, 72, 162, 520, 1514, 4312, 13242, 39088, 118586, 361712, 1103946, 3403624, 10513130, 32614696, 101530170, 316770752, 990771834, 3104283168, 9741133578, 30606719000, 96263812906, 303028237848, 954563802106, 3008665176560, 9487377712634, 29928407213328
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OFFSET
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0,3
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COMMENTS
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See A351322 for algorithm. The subsequence 1,8,162,... for 6 X 3n rectangles also has a depending recurrence with 11 parameters.
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
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (2,8,2,-43,-42,36,102,0,-44,-8,-8).
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FORMULA
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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).
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}.
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EXAMPLE
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For a 6 X 2 rectangle there are 4 tilings:
___ ___ ___ ___
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|___| |___| |___| |___|
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|___| |___| |___| |___|
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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