A054858 Number of basic blocks of size 5xn for tilings with square tiles of size up to 5 X 5.

1, 7, 13, 20, 35, 66, 118, 218, 402, 738, 1358, 2498, 4594, 8450, 15542, 28586, 52578, 96706, 177870, 327154, 601730, 1106754, 2035638, 3744122, 6886514, 12666274, 23296910, 42849698, 78812882, 144959490, 266622070

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

Table of n, a(n) for n=1..31.

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

Cf. A054857.

Sequence in context: A308200 A126621 A109331 * A269704 A157971 A135275

Adjacent sequences: A054855 A054856 A054857 * A054859 A054860 A054861

Keyword

easy,nonn

Author

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

Status

approved