|
%I #25 Aug 24 2020 22:42:20
%S 1,1,8,21,93,314,1213,4375,16334,59925,221799,817280,3018301,11134189,
%T 41096528,151643937,559640289,2065192514,7621289593,28124714395,
%U 103789150046,383013144129,1413437041011,5216013647648,19248692843977
%N Number of ways to tile a 5 X n area with 1 X 1 and 2 X 2 tiles.
%H S. Heubach, <a href="https://www.calstatela.edu/sites/default/files/users/u1231/Papers/cgtc30.pdf">Tiling an m-by-n area with squares of size up to k-by-k (m<=5)</a>, Congressus Numerantium 140 (1999), 43-64.
%H R. J. Mathar, <a href="http://arxiv.org/abs/1609.03964">Tiling nxm rectangles with 1 X 1 and s X s squares</a> arXiv:1609.03964 [math.CO], 2016.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,7,-2,-3).
%F a(n) = b(1)a(n-1)+b(2)a(n-2)+...+b(n)a(0), where a(0)=a(1)=1 and b(1)=1, b(2)=7, b(n)=F(n+1)of A000045 (Fibonacci numbers) for n>2.
%F a(n) = 2*a(n-1) + 7*a(n-2) - 2*a(n-3) - 3*a(n-4). - Keith Schneider (kschneid(AT)bulldog.unca.edu), Apr 02 2006
%F G.f.: (1-x-x^2)/(1-2*x-7*x^2+2*x^3+3*x^4). [_R. J. Mathar_, Nov 02 2008]
%e a(2)=8 as there is one tiling of a 5 X 2 area with only 1 X 1 tiles, 4 tilings with exactly one 2 X 2 tile and 3 tilings with exactly two 2 X 2 tiles.
%t f[{A_, B_}] := Module[{til = A, basic = B}, {Flatten[Append[til, ListConvolve[A, B]]], AppendTo[basic, 2 Fibonacci[Length[B] + 2]]}]; NumOfTilings[n_] := Nest[f, {{1, 1}, {1, 7}}, n - 2][[1]] NumOfTilings[30]
%Y Cf. A054854, A000045.
%Y Column k=5 of A245013.
%K easy,nonn
%O 0,3
%A Silvia Heubach (silvi(AT)cine.net), Apr 21 2000
|
