A045846 -id:A045846 - cp's OEIS frontend

A226997 Irregular triangle read by rows: T(n,k) is the number of distinct tilings by squares of an n X n square lattice that contain k nodes unconnected to any of their neighbors.

1, 1, 1, 1, 4, 0, 0, 1, 1, 9, 16, 8, 5, 0, 0, 0, 0, 1, 1, 16, 78, 140, 88, 44, 68, 32, 0, 4, 0, 0, 0, 0, 0, 0, 1, 1, 25, 228, 964, 2003, 2178, 1842, 1626, 725, 290, 376, 184, 140, 76, 4, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 36, 520, 3920, 16859, 42944, 67312

Offset: 1

Christopher Hunt Gribble, Jun 26 2013

The n-th row contains (n-1)^2 + 1 elements.

			For n = 3, there are 4 tilings that contain 1 isolated node, so T(3,1) = 4. A 2 X 2 square contains 1 isolated node.  Consider that each tiling is composed of ones and zeros where a one represents a node with one or more links to its neighbors and a zero represents a node with no links to its neighbors.  Then the 4 tilings are:
1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1
1 0 1 1    1 1 0 1    1 1 1 1    1 1 1 1
1 1 1 1    1 1 1 1    1 0 1 1    1 1 0 1
1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1
The irregular triangle begins:
\ k 0     1     2     3     4     5     6     7     8     9  ...
n
1   1
2   1     1
3   1     4     0     0     1
4   1     9    16     8     5     0     0     0     0     1
5   1    16    78   140    88    44    68    32     0     4  ...
6   1    25   228   964  2003  2178  1842  1626   725   290  ...
7   1    36   520  3920 16859 42944 67312 72980 69741 62952  ...

Alois P. Heinz, Rows n = 1..16, flattened (Rows n = 1..7 from Christopher Hunt Gribble)

Cf. A045846.

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+x^((i-k)^2)
          *b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)])
         od; expand(s)
      fi
    end:
T:= n-> (l-> seq(coeff(l,x,i), i=0..degree(l)))(b(n, [0$n])):
seq(T(n), n=1..9);  # Alois P. Heinz, Jun 27 2013

Sum_{k=0..(n-1)^2} T(n,k) = A045846(n).
From Christopher Hunt Gribble, Jul 02 2013: (Start)
It appears that:
T(n,1) = (n-1)^2, n>1 = A000290(n-1).
T(n,2) = (n-2)(n-3)(n^2+n-4)/2, n>2 = A061995(n-1).
T(n,3) = (n-2)(n-3)(n^4-n^3-23n^2+15n+140)/6, n>2 = A061996(n-1).
T(n,4) = (n^8 - 8n^7 - 26*n^6 + 340*n^5 - 105*n^4 - 4708*n^3 + 6814*n^2 + 20852*n - 40248)/24, n>3. (End)

A362143 Maximum number of ways in which a set of integer-sided squares can tile an n X n square.

Original entry on oeis.org

1, 1, 1, 4, 16, 140, 1987, 62266, 3899340, 508317004, 108388350636, 44608244030240, 39116330784279236

Offset: 0

Pontus von Brömssen, Apr 10 2023

Main diagonal of A362142.

Cf. A034295, A045846, A361217 (rectangular pieces).

A334617 a(n) is the number of ways to tile a size n staircase polyomino with staircase polyominoes.

Original entry on oeis.org

1, 2, 8, 57, 806, 20840, 1038266, 97115638, 17213517207, 5768580741287

Offset: 1

Peter Kagey, Sep 08 2020

A size-n staircase polynomo is a polyomino consisting of n left-aligned rows in increasing length of 1, 2, ..., n. Rotations of staircase polyominoes are also polyominoes.

			For n = 3 the a(3) = 8 tilings are:
+---+          +---+          +---+          +---+
|   |          |   |          |   |          |   |
+---+---+      +   +---+      +---+---+      +---+---+
|   |   |      |       |      |   |   |      |   |   |
+---+---+---+, +---+---+---+, +   +---+---+, +---+   +---+,
|   |   |   |  |   |   |   |  |       |   |  |   |       |
+---+---+---+  +---+---+---+  +---+---+---+  +---+---+---+
+---+          +---+          +---+          +---+
|   |          |   |          |   |          |   |
+---+---+      +---+---+      +---+---+      +   +---+
|       |      |       |      |   |   |      |       |
+---+   +---+, +   +---+---+, +---+   +---+, +       +---+.
|   |   |   |  |   |   |   |  |       |   |  |           |
+---+---+---+  +---+---+---+  +---+---+---+  +---+---+---+

Code Golf Stack Exchange user "Bubbler", Tiling a staircase with staircases.

Cf. A000105, A045846, A219924, A254414, A335547.

a(8) from Seiichi Manyama, Sep 09 2020

a(9)-a(10) from Bert Dobbelaere, Sep 12 2020

A347800 Number of tilings of an n X n square using integer-sided square tiles of area > 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 7, 1, 48, 50, 796, 777, 24823, 9315, 1501873, 2585314, 200614419, 382810931, 52597087873, 48712723680, 27115688491527

Offset: 0

Alois P. Heinz, Sep 14 2021

			a(4) = 2:
  ._._._._.   ._._._._.
  |       |   |   |   |
  |       |   |___|___|
  |       |   |   |   |
  |_______|   |___|___|   .

Main diagonal of A226206.

Cf. A045846.

a(n) = A226206(n,n).

A358715 a(n) is the number of distinct ways to cut an equilateral triangle with edges of size n into equilateral triangles with integer sides.

Original entry on oeis.org

1, 2, 5, 26, 220, 3622, 105859, 5677789, 553715341, 98404068313, 31850967186980, 18779046566454536, 20167518569123722322, 39451359692134386945019

Offset: 1

Craig Knecht and John Mason, Nov 28 2022

In other words, the number of equilateral triangular tilings of an equilateral triangle, where rotations and reflections are considered distinct.

			a(3)=5 because of:
    /\      /\      /\      /\      /\
   /  \    /\/\    /  \    /\/\    /\/\
  /    \  /  \/\  /\/\/\  /\/  \  /\/\/\

Cf. A045846, A097076, A358716.

Cf. A290820, A290821, A299705, A300001.

a(10)-a(14) from Walter Trump, Dec 03 2022

A380608 a(n) is the number of distinct ways to cut a hexagon with edges of size n into diamonds with integer sides.

Original entry on oeis.org

2, 37, 6330, 12807773

Offset: 0

Craig Knecht, Jan 28 2025

The number of ways to cut a diamond with edges of size n into diamonds with integer sides is A045846.

Craig Knecht, Diamond division of the hexagon.

Cf. A045846.

cp's OEIS Frontend

A226997 Irregular triangle read by rows: T(n,k) is the number of distinct tilings by squares of an n X n square lattice that contain k nodes unconnected to any of their neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A362143 Maximum number of ways in which a set of integer-sided squares can tile an n X n square.

Views

Author

Keywords

Crossrefs

A334617 a(n) is the number of ways to tile a size n staircase polyomino with staircase polyominoes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A347800 Number of tilings of an n X n square using integer-sided square tiles of area > 1.

Views

Author

Keywords

Examples

Crossrefs

Formula

A358715 a(n) is the number of distinct ways to cut an equilateral triangle with edges of size n into equilateral triangles with integer sides.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A380608 a(n) is the number of distinct ways to cut a hexagon with edges of size n into diamonds with integer sides.

Views

Author

Keywords

Comments

Links

Crossrefs