A054855 Number of ways to tile a 5 X n area with 1 X 1 and 2 X 2 tiles.

1, 1, 8, 21, 93, 314, 1213, 4375, 16334, 59925, 221799, 817280, 3018301, 11134189, 41096528, 151643937, 559640289, 2065192514, 7621289593, 28124714395, 103789150046, 383013144129, 1413437041011, 5216013647648, 19248692843977

Offset

0,3

Links

Table of n, a(n) for n=0..24.

S. Heubach, Tiling an m-by-n area with squares of size up to k-by-k (m<=5), Congressus Numerantium 140 (1999), 43-64.

R. J. Mathar, Tiling nxm rectangles with 1 X 1 and s X s squares arXiv:1609.03964 [math.CO], 2016.

Index entries for linear recurrences with constant coefficients, signature (2,7,-2,-3).

Formula

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.

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

G.f.: (1-x-x^2)/(1-2*x-7*x^2+2*x^3+3*x^4). [R. J. Mathar, Nov 02 2008]

Example

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.

Mathematica

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]

Crossrefs

Cf. A054854, A000045.

Column k=5 of A245013.

Sequence in context: A275185 A264238 A188700 * A220611 A297340 A220580

Adjacent sequences: A054852 A054853 A054854 * A054856 A054857 A054858

Keyword

easy,nonn

Author

Silvia Heubach (silvi(AT)cine.net), Apr 21 2000

Status

approved