OFFSET
0,3
COMMENTS
It is easy to see that the g.f. for indecomposable tilings, i.e. those that cannot be split vertically into smaller tilings, is g=z+4*z^2+2*z^3+z^4+2*z^3/(1-z); then G.f.=1/(1-g). - Emeric Deutsch, Oct 16 2006
LINKS
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.
Index entries for linear recurrences with constant coefficients, signature (2,3,0,-1,-1).
FORMULA
a(n) = a(n-1)+4*a(n-2)+4*a(n-3)+3*a(n-4)+2*( a(n-5)+a(n-6)+...+a(0)), a(0)=a(1)=1, a(2)=5, a(3)=13
a(n) = 2*a(n-1)+3*a(n-2)-a(n-4)-a(n-5). G.f.=(1-z)/((1+z)*(1-3*z+z^4)). - Emeric Deutsch, Oct 16 2006
EXAMPLE
a(2)=5 as there is one tiling of a 4 X 2 region with only 1 X 1 tiles, 3 tilings with exactly one 2 X 2 tile and 1 tiling with exactly two 2 X 2 tiles.
MAPLE
a[0]:=1: a[1]:=1: a[2]:=5: a[3]:=13: a[4]:=40: for n from 5 to 26 do a[n]:=2*a[n-1]+3*a[n-2]-a[n-4]-a[n-5] od: seq(a[n], n=0..26); # Emeric Deutsch, Oct 16 2006
MATHEMATICA
f[ A_ ] := Module[ {til = A, sum}, sum = 2* Apply[ Plus, Drop[ til, -4 ] ]; AppendTo[ til, A[ [ -1 ] ] + 4A[ [ -2 ] ] + 4A[ [ -3 ] ] + 3A[ [ -4 ] ] + sum ] ]; NumOfTilings[ n_ ] := Nest[ f, {1, 1, 5, 13}, n - 2 ]; NumOfTilings[ 30 ]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Silvia Heubach (silvi(AT)cine.net), Apr 21 2000
STATUS
approved