1,2

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).

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

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+2x^4-x^3+5x^2-x-10+2(-4x+5-5x^2)/(1-x-x^2-x^3). a(n) = 10*A000213(n)-8*A000073(n+1), n>5. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 02 2008]

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.

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 ]

Cf. A054857.

Sequence in context: A023984 A126621 A109331 this_sequence A135275 A013652 A099988

Adjacent sequences: A054855 A054856 A054857 this_sequence A054859 A054860 A054861

easy,nonn,new

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