OFFSET
1,2
COMMENTS
Basic blocks of size 5xn are tilings of a 5xn area that cannot be vertically split into two smaller tilings of size 5xk and 5x(n-k).
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 (1,1,1).
FORMULA
a(n) = a(n-1)+a(n-2)+a(n-3) for n>8, a(1)=1, a(2)=7, a(3)=13, a(4)=20, a(5)=35, a(6)=66, a(7)=218
G.f.: x^5+2*x^4-x^3+5*x^2-x-10+2*(-4*x+5-5*x^2)/(1-x-x^2-x^3). a(n) = 10*A000213(n)-8*A000073(n+1), n>5. [R. J. Mathar, Nov 02 2008]
EXAMPLE
a(3)=7 as the nature of basic blocks requires that the tiling cannot be split vertically into smaller tilings. Thus there needs to be one 2 X 2 tile whose lower left corner is in column 1 and one whose llc is in column 2. There are 7 ways to place these two 2 X 2 tiles.
MATHEMATICA
f[ {A_, B_} ] := Module[ {til = A, basic = B}, {Flatten[ Append[ til, ListConvolve[ A, B ] ]], AppendTo[ basic, B[[ -1 ]] + B[[ -2 ]] + B[[ -3 ] ]]} ]; NumOfBasicBlocks[ n_ ] := Nest[ f, {{1, 1, 8, 28, 117, 472, 1916, 7765}, {1, 7, 13, 20, 35, 66, 118, 218}}, n-2 ][[ 2 ]] NumOfBasicBlocks[ 30 ]
LinearRecurrence[{1, 1, 1}, {1, 7, 13, 20, 35, 66, 118, 218}, 40] (* Harvey P. Dale, Dec 06 2018 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Silvia Heubach (silvi(AT)cine.net), Apr 21 2000
STATUS
approved