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.

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A362142 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided squares can tile an n X k rectangle, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 3, 6, 16, 1, 1, 4, 12, 37, 140, 1, 1, 6, 24, 105, 454, 1987, 1, 1, 10, 40, 250, 1566, 9856, 62266, 1, 1, 15, 80, 726, 5670, 47394, 406168, 3899340, 1, 1, 21, 160, 1824, 18738, 223696, 2916492, 38322758, 508317004
Offset: 0

Views

Author

Pontus von Brömssen, Apr 10 2023

Keywords

Examples

			Triangle begins:
  n\k| 0  1  2  3   4    5     6      7       8
  ---+-----------------------------------------
  0  | 1
  1  | 1  1
  2  | 1  1  1
  3  | 1  1  2  4
  4  | 1  1  3  6  16
  5  | 1  1  4 12  37  140
  6  | 1  1  6 24 105  454  1987
  7  | 1  1 10 40 250 1566  9856  62266
  8  | 1  1 15 80 726 5670 47394 406168 3899340
A 5 X 4 rectangle can be tiled by 12 unit squares and 2 squares of side 2 in the following ways:
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |   |   |   |       |   |   |   |   |       |   |   |   |   |       |
  +---+---+---+---+   +       +---+---+   +---+       +---+   +---+---+       +
  |   |   |       |   |       |   |   |   |   |       |   |   |   |   |       |
  +---+---+       +   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |       |       |   |       |   |   |   |       |   |   |   |       |   |   |
  +       +---+---+   +       +---+---+   +       +---+---+   +       +---+---+
  |       |   |   |   |       |   |   |   |       |   |   |   |       |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
.
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |       |   |   |   |   |   |   |   |   |   |   |   |   |       |   |   |
  +---+       +---+   +---+---+---+---+   +---+---+---+---+   +       +---+---+
  |   |       |   |   |       |   |   |   |   |   |   |   |   |       |   |   |
  +---+---+---+---+   +       +---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |   |   |   |       |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |       |   |   |   |   |       |   |   |       |       |   |       |   |   |
  +       +---+---+   +---+       +---+   +       +       +   +       +---+---+
  |       |   |   |   |   |       |   |   |       |       |   |       |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
.
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |       |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+       +   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |       |   |   |       |   |   |   |   |   |   |   |   |   |       |
  +---+---+---+---+   +---+       +---+   +---+---+---+---+   +---+---+       +
  |   |   |   |   |   |   |       |   |   |       |       |   |       |       |
  +---+---+---+---+   +---+---+---+---+   +       +       +   +       +---+---+
  |       |   |   |   |   |       |   |   |       |       |   |       |   |   |
  +       +---+---+   +---+       +---+   +---+---+---+---+   +---+---+---+---+
  |       |   |   |   |   |       |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
.
  +---+---+---+---+
  |   |       |   |
  +---+       +---+
  |   |       |   |
  +---+---+---+---+
  |   |   |   |   |
  +---+---+---+---+
  |   |       |   |
  +---+       +---+
  |   |       |   |
  +---+---+---+---+
The first six of these have no symmetries, so they account for 4 tilings each. The next six have either a mirror symmetry or a rotational symmetry and account for 2 tilings each. The last has full symmetry and accounts for 1 tiling. In total there are 6*4+6*2+1 = 37 tilings. This is the maximum for a 5 X 4 rectangle, so T(5,4) = 37.

Links

Crossrefs

Main diagonal: A362143.
Columns: A000012 (k = 0,1), A073028 (k = 2), A362144 (k = 3), A362145 (k = 4), A362146 (k = 5).
Cf. A219924, A224697, A361216 (rectangular pieces).

A362261 Maximum number of ways in which a set of integer-sided squares can tile an n X 3 rectangle, up to rotations and reflections.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 12, 22, 40, 73, 146, 292, 560, 1120, 2532, 5040, 10080, 22176, 44352, 88704, 192272, 384384, 768768, 1647360, 3294720, 6589440, 14003120, 28006240, 56012480, 126028080, 266053680, 532107360, 1182438400, 2483130720, 4966261440, 10925775168
Offset: 0

Views

Author

Pontus von Brömssen, Apr 15 2023

Keywords

Crossrefs

Third column of A362258.
Cf. A359019, A361225 (rectangular pieces), A362144.

Programs

  • Python
    from math import comb
def F(i,j,k):
    # total number of tilings using i, j, and 2*j+3*k squares of side lengths 3, 2, and 1, respectively
    return comb(i+j+k,i)*comb(j+k,j)*2**j
def F0(i,j,k):
    # number of inequivalent tilings
    x = F(i,j,k)
    if j == 0: x += comb(i+k,i) # horizontal line of symmetry
    if i%2+j%2+k%2 <= 1: x += 2*F(i//2,j//2,k//2) # vertical line of symmetry or rotational symmetry
    return x//4
def A362261(n):
    return max(F0(i,j,n-3*i-2*j) for i in range(n//3+1) for j in range((n-3*i)//2+1))

Formula

a(n) >= A362144(n)/4.
Showing 1-2 of 2 results.

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