%I #26 Aug 24 2020 22:42:53
%S 1,1,8,28,117,472,1916,7765,31497,127707,517881,2100025,8515772,
%T 34532063,140030059,567832091,2302600696,9337214060,37863085664,
%U 153537580071,622606110920,2524713292359,10237896957896,41515420557135
%N Number of ways to tile a 5 X n region with square tiles of size up to 5 X 5.
%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.
%F a(n) = b(1)a(n-1)+b(2)a(n-2)+...+b(n)a(0), a(0)=a(1)=1, b(n) as defined in A054858.
%F a(n) = 2*a(n-1) +7*a(n-2) +6*a(n-3) -a(n-4) -5*a(n-5) -2*a(n-6) -3*a(n-7) -a(n-8). - _R. J. Mathar_, Nov 02 2008
%F G.f.: -(x^3+x^2+x-1)/(x^8+3*x^7+2*x^6+5*x^5+x^4-6*x^3-7*x^2-2*x+1). - _Colin Barker_, Jul 10 2012
%e a(2) = 8 as there is 1 tiling of a 5 X 2 region 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, B[ [ -1 ] ] + B[ [ -2 ] ] + B[ [ -3 ] ] ]} ]; NumOfTilings[ n_ ] := Nest[ f, {{1, 1, 8, 28, 117, 472, 1916, 7765}, {1, 7, 13, 20, 35, 66, 118, 218}}, n - 2 ][ [ 1 ] ] NumOfTilings[ 30 ]
%Y Cf. A002478, A054856, A054858, A226548.
%Y Column k=5 of A219924. - _Alois P. Heinz_, Dec 01 2012
%K nonn,easy
%O 0,3
%A Silvia Heubach (silvi(AT)cine.net), Apr 21 2000