A219924 -id:A219924 - cp's OEIS frontend

A045846 Number of distinct ways to cut an n X n square into squares with integer sides.

1, 1, 2, 6, 40, 472, 10668, 450924, 35863972, 5353011036, 1500957422222, 790347882174804, 781621363452405930, 1451740730942350766748, 5064070747064013556294032, 33176273260130056822126522884, 408199838581532754602910469192704

Offset: 0

Erich Friedman

			For n=3 the 6 dissections are: the full 3 X 3 square; 9 1 X 1 squares; one 2 X 2 square and five 1 X 1 squares (in 4 ways).

Andrew Gozzard and Max Ward, Table of n, a(n) for n = 0..25 (terms 0..20 from Steve Butler).
Steve Butler, Jason Ekstrand, Steven Osborne, Counting Tilings by Taking Walks in a Graph, A Project-Based Guide to Undergraduate Research in Mathematics, Birkhäuser, Cham (2020), see page 169.
N. J. A. Sloane, Illustration of the first five terms of A045846 and A224239, page 1 of 4 (Each dissection from A224239 is labeled with the number of its images under the symmetry group of the square. The sum of these numbers is A045846(n).)
N. J. A. Sloane, Illustration of the first five terms of A045846 and A224239, page 2 of 4 (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
N. J. A. Sloane, Illustration of the first five terms of A045846 and A224239, page 3 of 4 (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
N. J. A. Sloane, Illustration of the first five terms of A045846 and A224239, page 4 of 4 (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420 [math.CO], 2013-2014.

Diagonal of A219924. - Alois P. Heinz, Dec 01 2012
See A224239 for the number of inequivalent ways.
Cf. A034295, A063443, A211348, A226554.

b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then 0 elif n=0 or l=[] then 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; s:=0;
         for i from k to nops(l) while l[i]=0 do s:=s+
           b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)])
         od; s
      fi
    end:
a:= n-> b(n, [0$n]):
seq(a(n), n=0..11);  # Alois P. Heinz, Apr 15 2013

$RecursionLimit = 1000; b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which[ Max[l]>n, 0, n == 0 || l == {}, 1, Min[l]>0, t = Min[l]; b[n-t, l-t], True, For[k = 1, True, k++, If[l[[k]] == 0, Break[]]]; s=0; For[i=k, i <= Length[l] && l[[i]] == 0, i++, s = s + b[n, Join[l[[1 ;; k-1]], Table[1+i-k, {i-k+1}], l[[i+1 ;; Length[l]]]]]]; s]]; a[n_] := b[n, Array[0&, n]]; Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)

It appears lim n->oo a(n)*a(n-3)/(a(n-1)*a(n-2)) = 3.527... - Gerald McGarvey, May 03 2005
It appears that lim n->oo a(n)*a(n-2)/(a(n-1))^2 = 1.8781... - Christopher Hunt Gribble, Jun 21 2013
a(n) = (1/n^2) * Sum_{k=1..n} k^2 * A226936(n,k). - Alois P. Heinz, Jun 22 2013

More terms from Hugo van der Sanden, Nov 06 2000
a(14)-a(15) from Alois P. Heinz, Nov 30 2012
a(16) from Steve Butler, Mar 14 2014

A224239 Number of inequivalent ways to cut an n X n square into squares with integer sides.

Original entry on oeis.org

1, 2, 3, 13, 77, 1494, 56978, 4495023, 669203528, 187623057932, 98793520541768, 97702673827558670

Offset: 1

N. J. A. Sloane, Apr 15 2013

Similar to A045846, but now we do not regard dissections which differ by a rotation and/or reflection as distinct.

			For n=5, the illustrations (see links) show that the 77 solutions consist of:
4 dissections each with 1 image under the group of the square, for a total of 4,
2 dissections each with 2 images under the group of the square, totaling 4,
26 dissections each with 4 images under the group of the square, totaling 104, and
45 dissections each with 8 images under the group of the square, totaling 360,
for a grand total of 77 dissections with 472 images, agreeing with A045846(5) = 472.

Don Reble, C programs for A224239
Don Reble, Comments on the calculation of a(10)
N. J. A. Sloane, Illustration of the first five terms, page 1 of 4 (Each dissection is labeled with the number of its images under the symmetry group of the square. The sum of these numbers is A045846(n).)
N. J. A. Sloane, Illustration of the first five terms, page 2 of 4 (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
N. J. A. Sloane, Illustration of the first five terms, page 3 of 4 (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
N. J. A. Sloane, Illustration of the first five terms, page 4 of 4 (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, 2013; arXiv:1308.5420

Main diagonal of A227690.

Cf. A045846, A034295, A219924.

a(6)-a(10) from Don Reble, Apr 15 2013

a(11)-a(12) from Ed Wynn, 2013. - N. J. A. Sloane, Nov 29 2013

A113881 Table of smallest number of squares, T(m,n), needed to tile an m X n rectangle, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 3, 3, 4, 5, 2, 1, 2, 5, 6, 4, 4, 4, 4, 6, 7, 3, 4, 1, 4, 3, 7, 8, 5, 2, 5, 5, 2, 5, 8, 9, 4, 5, 3, 1, 3, 5, 4, 9, 10, 6, 5, 5, 5, 5, 5, 5, 6, 10, 11, 5, 3, 2, 5, 1, 5, 2, 3, 5, 11, 12, 7, 6, 6, 5, 5, 5, 5, 6, 6, 7, 12, 13, 6, 6, 4, 6, 4, 1, 4, 6, 4, 6, 6, 13, 14, 8, 4, 6, 2, 3, 7, 7, 3, 2, 6, 4, 8, 14

Offset: 1

Devin Kilminster (devin(AT)27720.net), Jan 27 2006

a(n) = A338573(n) for n <= 105, as stated by R. J. Mathar. These sequences are essentially different though, because a(13433) = T(67,98) = T(98,67) = a(13464), but A338573(13433) != A338573(13464). The relationship between the tiling problem and resistor networks is remarkable. There are explanations in M. Ortolano et al., 2013. - Rainer Rosenthal, Nov 09 2020

			T(n,n) = 1 (1 n X n square).
T(n,1) = n (n 1 X 1 squares).
T(6,7) = 6 (2 3 X 3, 1 4 X 4, 1 2 X 2, 2 1 X 1).
T(11,13) = 6 (1 7 X 7, 1 6 X 6, 1 5 X 5, 2 4 X 4 1 1 X 1).
Table T(m,n) begins:
:   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
:   2, 1, 3, 2, 4, 3, 5, 4, 6,  5, ...
:   3, 3, 1, 4, 4, 2, 5, 5, 3,  6, ...
:   4, 2, 4, 1, 5, 3, 5, 2, 6,  4, ...
:   5, 4, 4, 5, 1, 5, 5, 5, 6,  2, ...
:   6, 3, 2, 3, 5, 1, 5, 4, 3,  4, ...
:   7, 5, 5, 5, 5, 5, 1, 7, 6,  6, ...
:   8, 4, 5, 2, 5, 4, 7, 1, 7,  5, ...
:   9, 6, 3, 6, 6, 3, 6, 7, 1,  6, ...
:  10, 5, 6, 4, 2, 4, 6, 5, 6,  1, ...

Alois P. Heinz, Antidiagonals n = 1..350, flattened (using data from A219158)
Bertram Felgenhauer, Filling rectangles with integer-sided squares
Richard J. Kenyon, Tiling a rectangle with the fewest squares, Combin. Theory Ser. A 76 (1996), no. 2, 272-291.
M. Ortolano, M. Abrate, and L. Callegaro, On the synthesis of Quantum Hall Array Resistance Standards, arXiv preprint arXiv:1311.0756 [physics.ins-det], 2013.
Mark Walters, Rectangles as sums of squares, Discrete Math. 309 (2009), no. 9, 2913-2921.

Rows (or columns) m=1-10 give: A001477, A030451, A226576, A226577, A226578, A226579, A226580, A226581, A226582, A226583.

Cf. A219158, A219924, A226545, A338573.

(* *** Warning *** This empirical toy-program is based on the greedy algorithm. Its output was only verified for n+k <= 32. Any use outside this domain might produce only upper bounds instead of minimums. *)
nmax = 31; Clear[T];
Tmin[n_, k_] := Table[{1 + T[ c, k - c] + T[n - c, k], 1 + T[n, k - c] + T[n - c, c]}, {c, 1, k - 1}] // Flatten // Min;
Tmin2[n_, k_] := Module[{n1, n2, k1, k2}, 1 + T[n2, k1 + 1] + T[n - n1, k2] + T[n - n2, k1] + T[n1, k - k1] /. {Reduce[1 <= n1 <= n - 1 && 1 <= n2 <= n - 1 && 1 <= k1 <= k - 1 && 1 <= k2 <= k - 1 && n1 + 1 + n2 == n && k1 + 1 + k2 == k, Integers] // ToRules} // Min];
T[n_, n_] = 1;
T[n_, 1] := n;
T[1, k_] := k;
T[n_, k_ /; k > 1] /; n > k && Divisible[n, k] := n/k;
T[n_, k_ /; k > 1] /; n > k := T[n, k] = If[k >= 5 && n >= 6 && n - k <= 3, Min[Tmin[n, k], Tmin2[n, k], T[k, n - k] + 1], T[k, n - k] + 1];
T[n_, k_ /; k > 1] /; n < k := T[n, k] = T[k, n];
Table[T[n - k + 1, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 11 2016, checked against first 496 terms of the b-file *)

A054856 Number of ways to tile a 4 X n region with 1 X 1, 2 X 2, 3 X 3 and 4 X 4 tiles.

Original entry on oeis.org

1, 1, 5, 13, 40, 117, 348, 1029, 3049, 9028, 26738, 79183, 234502, 694476, 2056692, 6090891, 18038173, 53420041, 158203433, 468519406, 1387520047, 4109140098, 12169216863, 36039131181, 106729873498, 316080480394, 936072224321

Offset: 0

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

It is easy to see that the g.f. for indecomposable tilings, i.e. those that cannot be split vertically into smaller tilings, is g=z+4*z^2+2*z^3+z^4+2*z^3/(1-z); then G.f.=1/(1-g). - Emeric Deutsch, Oct 16 2006

			a(2)=5 as there is one tiling of a 4 X 2 region with only 1 X 1 tiles, 3 tilings with exactly one 2 X 2 tile and 1 tiling with exactly 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 (2,3,0,-1,-1).

Cf. A002478, A054857, A226547.

Column k=4 of A219924. - Alois P. Heinz, Dec 01 2012

a[0]:=1: a[1]:=1: a[2]:=5: a[3]:=13: a[4]:=40: for n from 5 to 26 do a[n]:=2*a[n-1]+3*a[n-2]-a[n-4]-a[n-5] od: seq(a[n],n=0..26); # Emeric Deutsch, Oct 16 2006

f[ A_ ] := Module[ {til = A, sum}, sum = 2* Apply[ Plus, Drop[ til, -4 ] ]; AppendTo[ til, A[ [ -1 ] ] + 4A[ [ -2 ] ] + 4A[ [ -3 ] ] + 3A[ [ -4 ] ] + sum ] ]; NumOfTilings[ n_ ] := Nest[ f, {1, 1, 5, 13}, n - 2 ]; NumOfTilings[ 30 ]

a(n) = a(n-1)+4*a(n-2)+4*a(n-3)+3*a(n-4)+2*( a(n-5)+a(n-6)+...+a(0)), a(0)=a(1)=1, a(2)=5, a(3)=13

a(n) = 2*a(n-1)+3*a(n-2)-a(n-4)-a(n-5). G.f.=(1-z)/((1+z)*(1-3*z+z^4)). - Emeric Deutsch, Oct 16 2006

A227690 Number A(n,k) of tilings of a k X n rectangle using integer-sided square tiles reduced for symmetry; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 3, 4, 1, 1, 1, 1, 5, 6, 6, 5, 1, 1, 1, 1, 9, 10, 13, 10, 9, 1, 1, 1, 1, 12, 21, 39, 39, 21, 12, 1, 1, 1, 1, 21, 39, 115, 77, 115, 39, 21, 1, 1, 1, 1, 30, 82, 295, 521, 521, 295, 82, 30, 1, 1

Offset: 0

Christopher Hunt Gribble, Jul 19 2013

			Square array A(n,k) begins:
  1, 1,  1,  1,   1,    1,     1,      1,       1, ...
  1, 1,  1,  1,   1,    1,     1,      1,       1, ...
  1, 1,  2,  2,   4,    5,     9,     12,      21, ...
  1, 1,  2,  3,   6,   10,    21,     39,      82, ...
  1, 1,  4,  6,  13,   39,   115,    295,     861, ...
  1, 1,  5, 10,  39,   77,   521,   1985,    8038, ...
  1, 1,  9, 21, 115,  521,  1494,  15129,   83609, ...
  1, 1, 12, 39, 295, 1985, 15129,  56978,  861159, ...
  1, 1, 21, 82, 861, 8038, 83609, 861159, 4495023, ...
...
A(4,3) = 6 because there are 6 ways to tile a 3 X 4 rectangle by subsquares, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct:
   ._____ _.    ._______.    ._______.
   |     |_|    |   |   |    |   |_|_|
   |     |_|    |___|_ _|    |___|   |
   |_____|_|    |_|_|_|_|    |_|_|___|
   ._______.    ._______.    ._______.
   |   |_|_|    |_|   |_|    |_|_|_|_|
   |___|_|_|    |_|___|_|    |_|_|_|_|
   |_|_|_|_|    |_|_|_|_|    |_|_|_|_|

Christopher Hunt Gribble, Antidiagonals n = 0..15, flattened
Christopher Hunt Gribble, C++ program

Main diagonal: A224239.
Columns 1-10: A000012, A001224, A359019, A359020, A359021, A359022, A359023, A359024, A359025, A359026.
Cf. A219924, A224697.

A054857 Number of ways to tile a 5 X n region with square tiles of size up to 5 X 5.

Original entry on oeis.org

1, 1, 8, 28, 117, 472, 1916, 7765, 31497, 127707, 517881, 2100025, 8515772, 34532063, 140030059, 567832091, 2302600696, 9337214060, 37863085664, 153537580071, 622606110920, 2524713292359, 10237896957896, 41515420557135

Offset: 0

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

			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.

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.

Cf. A002478, A054856, A054858, A226548.

Column k=5 of A219924. - Alois P. Heinz, Dec 01 2012

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 ]

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.
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
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

A226545 Number A(n,k) of squares in all tilings of a k X n rectangle using integer-sided square tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 5, 3, 0, 0, 4, 12, 12, 4, 0, 0, 5, 25, 34, 25, 5, 0, 0, 6, 50, 98, 98, 50, 6, 0, 0, 7, 96, 256, 386, 256, 96, 7, 0, 0, 8, 180, 654, 1402, 1402, 654, 180, 8, 0, 0, 9, 331, 1625, 4938, 6940, 4938, 1625, 331, 9, 0

Offset: 0

Alois P. Heinz, Jun 10 2013

			A(3,3) = 1 + 6 + 6 + 6 + 6 + 9 = 34:
  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
  |     |  |   |_|  |_|   |  |_|_|_|  |_|_|_|  |_|_|_|
  |     |  |___|_|  |_|___|  |_|   |  |   |_|  |_|_|_|
  |_____|  |_|_|_|  |_|_|_|  |_|___|  |___|_|  |_|_|_|
Square array A(n,k) begins:
  0, 0,   0,    0,     0,      0,       0,        0, ...
  0, 1,   2,    3,     4,      5,       6,        7, ...
  0, 2,   5,   12,    25,     50,      96,      180, ...
  0, 3,  12,   34,    98,    256,     654,     1625, ...
  0, 4,  25,   98,   386,   1402,    4938,    16936, ...
  0, 5,  50,  256,  1402,   6940,   33502,   157279, ...
  0, 6,  96,  654,  4938,  33502,  221672,  1426734, ...
  0, 7, 180, 1625, 16936, 157279, 1426734, 12582472, ...

Alois P. Heinz, Antidiagonals n = 0..30, flattened

Columns (or rows) k=0-10 give: A000004, A001477, A067331(n-1) for n>0, A226546, A226547, A226548, A226549, A226550, A226551, A226552, A226553.
Main diagonal gives A226554.
Cf. A113881, A219924.

b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then [0,0] elif n=0 or l=[] then [1,0]
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; s:=[0$2];
         for i from k to nops(l) while l[i]=0 do s:=s+(h->h+[0, h[1]])
           (b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
         od; s
      fi
    end:
A:= (n, k)-> `if`(n>=k, b(n, [0$k]), b(k, [0$n]))[2]:
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which[Max[l] > n, {0, 0}, n == 0 || l == {}, {1, 0}, Min[l] > 0, t=Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s={0, 0}; For[i=k, i <= Length[l] && l[[i]] == 0, i++, s = s + Function[h, h+{0, h[[1]]}][b[n, Join[l[[1 ;; k-1]], Table[1+i-k, {j, k, i}], l[[i+1 ;; -1]]]]] ]; s]]; a[n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]][[2]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

A219925 Number of tilings of a 6 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 13, 60, 348, 1916, 10668, 59257, 329350, 1830234, 10171315, 56525022, 314128014, 1745708992, 9701463927, 53914132251, 299618062228, 1665073290365, 9253344266757, 51423790446062, 285778433090830, 1588162056821687, 8825923956549044, 49048479247236561

Offset: 0

Alois P. Heinz, Dec 01 2012

			a(2) = 13, because there are 13 tilings of a 6 X 2 rectangle using integer-sided square tiles:
._._.  .___.  ._._.  ._._.  ._._.  ._._.
|_|_|  |   |  |_|_|  |_|_|  |_|_|  |_|_|
|_|_|  |___|  |   |  |_|_|  |_|_|  |_|_|
|_|_|  |_|_|  |___|  |   |  |_|_|  |_|_|
|_|_|  |_|_|  |_|_|  |___|  |   |  |_|_|
|_|_|  |_|_|  |_|_|  |_|_|  |___|  |   |
|_|_|  |_|_|  |_|_|  |_|_|  |_|_|  |___|
.___.  .___.  .___.  ._._.  ._._.  ._._.  .___.
|   |  |   |  |   |  |_|_|  |_|_|  |_|_|  |   |
|___|  |___|  |___|  |   |  |   |  |_|_|  |___|
|   |  |_|_|  |_|_|  |___|  |___|  |   |  |   |
|___|  |   |  |_|_|  |   |  |_|_|  |___|  |___|
|_|_|  |___|  |   |  |___|  |   |  |   |  |   |
|_|_|  |_|_|  |___|  |_|_|  |___|  |___|  |___|

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=6 of A219924.

Cf. A226549.

gf:= -(2*x^9 +3*x^8 +2*x^7 -3*x^6 -7*x^5 -4*x^4 -3*x^3 +5*x^2 +2*x -1) / (2*x^15 +7*x^14 +12*x^13 +6*x^12 -18*x^11 -13*x^10 -8*x^9 -27*x^8 -32*x^7 +x^6 +40*x^5 +34*x^4 -3*x^3 -15*x^2 -3*x +1):
a:= n-> coeff (series (gf, x, n+1), x, n):
seq (a(n), n=0..40);

G.f.: see Maple program.

A219926 Number of tilings of a 7 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 21, 129, 1029, 7765, 59257, 450924, 3435392, 26160354, 199243634, 1517411733, 11556549312, 88013947545, 670309228276, 5105035683160, 38879655193542, 296105186372225, 2255119850966932, 17174861374796123, 130802743517191075, 996186073044886758

Offset: 0

Alois P. Heinz, Dec 01 2012

			a(2) = 21, because there are 21 tilings of a 7 X 2 rectangle using integer-sided square tiles:
._._. .___. ._._. ._._. ._._. ._._. ._._. .___. .___. .___. .___.
|_|_| |   | |_|_| |_|_| |_|_| |_|_| |_|_| |   | |   | |   | |   |
|_|_| |___| |   | |_|_| |_|_| |_|_| |_|_| |___| |___| |___| |___|
|_|_| |_|_| |___| |   | |_|_| |_|_| |_|_| |   | |_|_| |_|_| |_|_|
|_|_| |_|_| |_|_| |___| |   | |_|_| |_|_| |___| |   | |_|_| |_|_|
|_|_| |_|_| |_|_| |_|_| |___| |   | |_|_| |_|_| |___| |   | |_|_|
|_|_| |_|_| |_|_| |_|_| |_|_| |___| |   | |_|_| |_|_| |___| |   |
|_|_| |_|_| |_|_| |_|_| |_|_| |_|_| |___| |_|_| |_|_| |_|_| |___|
._._. ._._. ._._. ._._. ._._. ._._. .___. .___. .___. ._._.
|_|_| |_|_| |_|_| |_|_| |_|_| |_|_| |   | |   | |   | |_|_|
|   | |   | |   | |_|_| |_|_| |_|_| |___| |___| |___| |   |
|___| |___| |___| |   | |   | |_|_| |   | |   | |_|_| |___|
|   | |_|_| |_|_| |___| |___| |   | |___| |___| |   | |   |
|___| |   | |_|_| |   | |_|_| |___| |   | |_|_| |___| |___|
|_|_| |___| |   | |___| |   | |   | |___| |   | |   | |   |
|_|_| |_|_| |___| |_|_| |___| |___| |_|_| |___| |___| |___|

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=7 of A219924.

Cf. A226550.

gf:= -(6*x^18 -x^17 -9*x^16 +13*x^15 +20*x^14 -35*x^13 -47*x^12 -76*x^11 -145*x^10 -127*x^9 -8*x^8 +64*x^7 +96*x^6 +68*x^5 +7*x^4 -10*x^3 -13*x^2 -2*x +1) / (6*x^25 +11*x^24 -9*x^23 -10*x^22 +39*x^21 +12*x^20 -70*x^19 -281*x^18 -403*x^17 -110*x^16 -118*x^15 -790*x^14 -179*x^13 +466*x^12 +327*x^11 +669*x^10 +1028*x^9 +231*x^8 -45*x^7 -284*x^6 -273*x^5 -61*x^4 +45*x^3 +31*x^2 +3*x -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);

G.f.: see Maple program.

A219927 Number of tilings of an 8 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 34, 277, 3049, 31497, 329350, 3435392, 35863972, 374285478, 3906605183, 40773605243, 425562898029, 4441677458152, 46358636450427, 483853831650209, 5050074056261222, 52708577944998395, 550129399697072615, 5741804607960538038, 59928300863912394900

Offset: 0

Alois P. Heinz, Dec 01 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=8 of A219924.

Cf. A226551.

gf:= -(60*x^40 +136*x^39 -321*x^38 -1038*x^37 -2045*x^36 +2501*x^35 +4393*x^34 +7347*x^33 +4372*x^32 -4825*x^31 -13838*x^30 -19585*x^29 -9331*x^28 -40213*x^27 +19891*x^26 +57417*x^25 +68058*x^24 +10427*x^23 -8789*x^22 +6040*x^21 -76684*x^20 -81024*x^19 -16484*x^18 +11908*x^17 +42083*x^16 +63711*x^15 +19938*x^14 -2290*x^13 -18240*x^12 -18560*x^11 -7633*x^10 +291*x^9 +4194*x^8 +2502*x^7 +378*x^6 -361*x^5 -240*x^4 -33*x^3 +27*x^2 +5*x -1) /
(60*x^48 +256*x^47 +35*x^46 -1488*x^45 -4435*x^44 -2543*x^43 +7032*x^42 +16610*x^41 +23043*x^40 +18924*x^39 +3186*x^38 -57091*x^37 -115830*x^36 -141242*x^35 +18849*x^34 +39846*x^33 +240064*x^32 +433164*x^31 +162501*x^30 -692061*x^29 -641988*x^28 +446013*x^27 +530385*x^26 +657974*x^25 -654746*x^24 -708014*x^23 -43614*x^22 -370550*x^21 +356235*x^20 +824516*x^19 +224413*x^18 -94736*x^17 -143852*x^16 -344353*x^15 -110166*x^14 +15107*x^13 +55317*x^12 +51581*x^11 +29259*x^10 +6818*x^9 -5977*x^8 -8807*x^7 -2453*x^6 +1175*x^5 +708*x^4 +15*x^3 -55*x^2 -6*x +1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);

G.f.: see Maple program.

cp's OEIS Frontend

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