cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Views

Author

Christopher Hunt Gribble, Jul 19 2013

Keywords

Examples

			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:
   ._____ _.    ._______.    ._______.
   |     |_|    |   |   |    |   |_|_|
   |     |_|    |___|_ _|    |___|   |
   |_____|_|    |_|_|_|_|    |_|_|___|
   ._______.    ._______.    ._______.
   |   |_|_|    |_|   |_|    |_|_|_|_|
   |___|_|_|    |_|___|_|    |_|_|_|_|
   |_|_|_|_|    |_|_|_|_|    |_|_|_|_|

Links

Crossrefs

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

A359019 Number of inequivalent tilings of a 3 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 21, 39, 82, 163, 347, 717, 1533, 3232, 6927, 14748, 31645, 67690, 145322, 311535, 668997, 1435645, 3083301, 6619842, 14218066, 30533005, 65580338, 140847132, 302522253, 649759735, 1395611508, 2997573501, 6438470626, 13829057884, 29703388721, 63799607283, 137035047576, 294336860797, 632205714741
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Examples

			a(4) is 6 because of:
  +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
  | | | | |     | |   | | |   | | |   | | | | | |
  +-+-+-+ +     + +   +-+ +   +-+ +   +-+ +-+-+-+
  | | | | |     | |   | | |   | | |   | | |   | |
  +-+-+-+ +     + +-+-+-+ +-+-+-+ +-+-+-+ +   +-+
  | | | | |     | |   | | | |   | | | | | |   | |
  +-+-+-+ +-+-+-+ +   +-+ +-+   + +-+-+-+ +-+-+-+
  | | | | |     | |   | | | |   | | | | | | | | |
  +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+

Links

Crossrefs

Column k = 3 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.
Cf. A000930.

Formula

For n <= 1, a(n)=1;
otherwise for odd n > 1, a(n)=(A002478(n) + A000930(n) + 2 * A002478((n - 1) / 2) + 2 * A002478((n - 3) / 2)) / 4
and for even n, a(n)=(A002478(n) + A000930(n) + 2 * A002478((n - 2) / 2) + 2 * A002478(n / 2)) / 4
Alternatively, from Walter Trump:
For n <= 1, a(n)=1;
otherwise for odd n > 1, a(n)=(A000930(2n) + A000930(n) + 2 * A000930(n - 1) + 2 * A000930(n - 3)) / 4
and for even n, a(n)=(A000930(2n) + 2 * A000930(n - 2) + 3 * A000930(n)) / 4

A359020 Number of inequivalent tilings of a 4 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 4, 6, 13, 39, 115, 295, 861, 2403, 7048, 20377, 60008, 175978, 519589, 1532455, 4531277, 13395656, 39639758, 117301153, 347248981, 1028011708, 3043852214, 9012879842, 26689014028, 79033362580, 234045889421, 693101137571, 2052569508948
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Examples

			a(3) is 6 because of:
  +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
  | | | | |     | |   | | |   | | |   | | | | | |
  +-+-+-+ +     + +   +-+ +   +-+ +   +-+ +-+-+-+
  | | | | |     | |   | | |   | | |   | | |   | |
  +-+-+-+ +     + +-+-+-+ +-+-+-+ +-+-+-+ +   +-+
  | | | | |     | |   | | | |   | | | | | |   | |
  +-+-+-+ +-+-+-+ +   +-+ +-+   + +-+-+-+ +-+-+-+
  | | | | |     | |   | | | |   | | | | | | | | |
  +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+

Links

Crossrefs

Column k = 4 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.

Formula

For even n > 4
a(n) = (A054856(n) + compo(n) + 4 * A054856((n - 2) / 2) +
2 * A054856((n - 4) / 2) + 2 * A054856(n / 2) +
2 * Sum_{k=0..(n - 2) / 2} (A054856(k))) / 4
For odd n > 4
a(n) = (A054856(n) + compo(n) + 2 * A054856((n - 3) / 2) +
2 * A054856((n - 1) / 2) + 2 * Sum_ {k=0..(n - 3) / 2} (A054856(k))) / 4
Where compo(n) is the number of distinct compositions of n as a sum of 1, 2, (1+1) and 4.

A359022 Number of inequivalent tilings of a 6 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 9, 21, 115, 521, 1494, 15129, 83609, 459957, 2551794, 14150081, 78597739
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Crossrefs

Column k = 6 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.

A359023 Number of inequivalent tilings of a 7 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 12, 39, 295, 1985, 15129, 56978, 861159, 6542578, 49828415
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Crossrefs

Column k = 7 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.

A359024 Number of inequivalent tilings of an 8 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 21, 82, 861, 8038, 83609, 861159, 4495023
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Crossrefs

Column k = 8 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.

A359025 Number of inequivalent tilings of a 9 X n rectangle using integer-sided square tiles.

Original entry on oeis.org

1, 1, 30, 163, 2403, 32097, 459957, 6542578, 93604244
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Crossrefs

Column k = 9 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.

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

Original entry on oeis.org

1, 1, 51, 347, 7048, 130125, 2551794, 49828415
Offset: 0

Views

Author

John Mason, Dec 12 2022

Keywords

Crossrefs

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.

A362263 Maximum number of ways in which a set of integer-sided squares can tile an n X 5 rectangle, up to rotations and reflections.

Original entry on oeis.org

1, 1, 2, 4, 13, 20, 125, 403, 1438, 4718, 17700, 65811, 266345, 1036625, 3817494, 15399048
Offset: 0

Views

Author

Pontus von Brömssen, Apr 15 2023

Keywords

Crossrefs

Fifth column of A362258.
Cf. A359021, A362146.

Formula

a(n) >= A362146(n)/4 for n != 5.
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.