A054857 -id:A054857 - cp's OEIS frontend

A359026 Number of inequivalent tilings of a 10 X n rectangle using integer-sided square tiles.

1, 1, 51, 347, 7048, 130125, 2551794, 49828415

Offset: 0

John Mason, Dec 12 2022

Column k = 10 of A227690.
Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
Fixed: A000045, A002478, A054856, A054857, A219925, A219926, A219927, A219928, A219929.
Free: A001224, A359019, A359020, A359021, A359022, A359023, A359024, A359025, A359026.

A226548 Number of squares in all tilings of a 5 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

0, 5, 50, 256, 1402, 6940, 33502, 157279, 725080, 3293575, 14789640, 65785991, 290341336, 1272947979, 5549535668, 24075671345, 104002776564, 447586575256, 1919810662952, 8210002538205, 35015688030978, 148980204943747, 632466650426456, 2679623444793841

Offset: 0

Alois P. Heinz, Jun 10 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=5 of A226545.

Cf. A054857.

G.f.: (x^10 +12*x^9 +28*x^8 +54*x^7 +33*x^6 +18*x^5 -53*x^4 -42*x^3 +6*x^2 +30*x+5)*x / (x^8 +3*x^7 +2*x^6 +5*x^5 +x^4 -6*x^3 -7*x^2 -2*x+1)^2.

A362146 Maximum number of ways in which a set of integer-sided squares can tile an n X 5 rectangle.

Original entry on oeis.org

1, 1, 4, 12, 37, 140, 454, 1566, 5670, 18738, 70800, 263002, 1065240, 4146500, 15269976, 61593260, 233206636, 879932068, 3442083056, 12794936028, 49150133908, 188187379818, 690417160564, 2688772826468, 10355819787218, 42350139855568, 167122149149460

Offset: 0

Pontus von Brömssen, Apr 10 2023

nonn

Pontus von Brömssen, Table of n, a(n) for n = 0..150

Fifth column of A362142.

Cf. A054857, A187753.

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

Original entry on oeis.org

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

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

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).

			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.

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).

Cf. A054857.

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 *)

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]

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A359026 Number of inequivalent tilings of a 10 X n rectangle using integer-sided square tiles.

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A226548 Number of squares in all tilings of a 5 X n rectangle using integer-sided square tiles.

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A362146 Maximum number of ways in which a set of integer-sided squares can tile an n X 5 rectangle.

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

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