A227959 Number of tilings using monominoes and L-trominoes in 2 X n chessboard, such that three monominoes cannot occur together in shape of L-tromino.
1, 1, 4, 6, 20, 38, 104, 220, 556, 1244, 3024, 6944, 16576, 38536, 91216, 213280, 502864, 1178928, 2774592, 6512864, 15315072, 35969952, 84550912, 198634048, 466825152, 1096838208, 2577550336, 6056474880, 14232064256, 33441977216, 78583660288, 184655188480
Offset: 0
Links
- Gopinath A. R., Table of n, a(n) for n = 0..200
- Calvin Lin, Tiling, Discrete Mathematics Problem on Linear Recurrence Relations, Brilliant.
- Index entries for linear recurrences with constant coefficients, signature (0,4,2,2,4).
Crossrefs
Cf. A127864.
Programs
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Magma
I:=[1,1,4,6,20]; [n le 5 select I[n] else 4*Self(n-2)+2*Self(n-3)+ 2*Self(n-4)+4*Self(n-5): n in [1..35]]; // Vincenzo Librandi, Apr 30 2018
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Mathematica
LinearRecurrence[{0, 4, 2, 2, 4}, {1, 1, 4, 6, 20}, 33] (* or *) CoefficientList[Series[(1 + x)/(1 - 4 x^2 - 2 x^3 - 2 x^4 - 4 x^5), {x, 0, 33}], x] (* Vincenzo Librandi, Apr 30 2018 *) -
PARI
Vec( (1+x)/(1-4*x^2-2*x^3-2*x^4-4*x^5) +O(x^66) ) \\ Joerg Arndt, Aug 07 2013
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Sage
fx = (1+x)/(1-4*x^2-2*x^3-2*x^4-4*x^5) fxt = taylor(fx,x,0,50) for i in range(51): print(i, fxt.coefficient(x,i))
Formula
a(n) = 4*a(n-2) + 2*a(n-3) + 2*a(n-4) + 4*a(n-5), with a(0)=1, a(1)=1, a(2)=4, a(3)=6, and a(4)=20.
G.f.: (1+x)/(1-4*x^2-2*x^3-2*x^4-4*x^5).
Asymptotic formula: a(n) ~ 0.581189405182598 * 2.3498153157195^n.