A270063
Number of tilings of a 3 X n rectangle using monominoes and trominoes of any shape.
Original entry on oeis.org
1, 2, 14, 93, 590, 3710, 23509, 148796, 940916, 5952243, 37655502, 238204978, 1506870209, 9532433288, 60301864000, 381467462217, 2413150656124, 15265511608858, 96569119066667, 610893045560410, 3864489168524208, 24446630393071735, 154648573589573678
Offset: 0
A335747
Number of ways to tile vertically-fault-free 3 X n strip with squares and dominoes.
Original entry on oeis.org
1, 3, 13, 26, 66, 154, 380, 904, 2204, 5286, 12818, 30854, 74636, 179948, 434820, 1049122, 2533818, 6115538, 14766868, 35646080, 86064196, 207766110, 501609946, 1210964110, 2923573588, 7058053972, 17039774268
Offset: 0
a(2) = 13 thanks to these thirteen vertically-fault-free tilings of a 3 X 2 strip:
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I:=[26, 66, 154, 380]; [1,3,13] cat [n le 4 select I[n] else Self(n-1) +4*Self(n-2) -Self(n-3) -Self(n-4): n in [1..40]]; // G. C. Greubel, Jan 15 2022
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CoefficientList[Series[(1+2x+6x^2+2x^3-8x^4+x^6)/((1+x-x^2)(1-2x-x^2)), {x, 0, 26}], x] (* Michael De Vlieger, Jul 03 2020 *)
LinearRecurrence[{1,4,-1,-1}, {1,3,13,26,66,154,380}, 40] (* G. C. Greubel, Jan 15 2022 *)
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def P(n): return lucas_number1(n,2,-1)
def A335747(n): return (1/3)*(-9*bool(n==0) - 3*bool(n==1) + 3*bool(n==2) + 2*(3*P(n+1) + 2*P(n-1)) + 2*(-1)^n*fibonacci(n-1))
[A335747(n) for n in (0..40)] # G. C. Greubel, Jan 15 2022
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