A335967 -id:A335967 - cp's OEIS frontend

A337131 Row lengths of irregular triangle A335967.

1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 2, 1, 2, 1, 1, 1, 1, 2, 1, 4, 8, 2, 2, 1, 1, 4, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 4, 1, 1, 4, 4, 16, 8, 2, 4, 2, 2, 1, 1, 2, 1, 4, 8, 2, 2, 1, 1, 4, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 4, 1, 1, 2, 2, 8, 4, 1, 2, 1, 1, 4, 4, 8, 4, 16, 32, 8

Offset: 1

Rémy Sigrist, Sep 14 2020

nonn

All terms are powers of 2.

			For n = 13, the binary representation of 13 is "1101", so we consider the tilings of a size 4 staircase polyomino whose base has the following shape:
      .....
      .   .
      .   .....
      .       .
      +---+   .....
      |   |       .
      |   +---+---+---+
      | 1   1 | 0 | 1 |
      +-------+---+---+
There are two possible penultimate rows:
      .....              .....
      .   .              .   .
      .   .....          .   .....
      .   |   .          .       .
      +---+   +---+      +---+---+---+
      | 1 | 0   0 |      | 1 | 0 | 1 |
      |   +---+---+---+  |   +---+---+---+
      |       |   |   |  |       |   |   |
      +-------+---+---+, +-------+---+---+
so a(13) = 2.

Rémy Sigrist, PARI program for A337131

Cf. A000975, A335967.

PARI
```
\\ See Links section.
```

a(2^k-1) = 1 for any k >= 0.
a(2^k) = 1 for any k >= 0.
a(A000975(k)) = 2^(k-2) for any k >= 2.

A335547 a(n) is the number of ways to tile a size n staircase polyomino with staircase polyominoes in the same direction as the size n staircase polyomino.

Original entry on oeis.org

1, 2, 5, 18, 94, 709, 7710, 120882, 2732104, 89015152, 4180822859, 283067837700, 27628050712667, 3887236104777699, 788428930992492718, 230523466443694083587, 97162501670167108808501, 59035492675117768533460333, 51708108674446390274283614578, 65288256486029805607741923173692

Offset: 1

Seiichi Manyama, Sep 12 2020

nonn

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

			For n = 3 the a(3) = 5 tilings are:
+---+          +---+          +---+          +---+
|   |          |   |          |   |          |   |
+---+---+      +   +---+      +---+---+      +---+---+
|   |   |      |       |      |   |   |      |   |   |
+---+---+---+, +---+---+---+, +   +---+---+, +---+   +---+,
|   |   |   |  |   |   |   |  |       |   |  |   |       |
+---+---+---+  +---+---+---+  +---+---+---+  +---+---+---+
+---+
|   |
+   +---+
|       |
+       +---+.
|           |
+---+---+---+
For n = 4 the a(4) = 5+5+3+3+2 = 18 tilings are:
+---+              +---+              +---+
|   |              |   |              |   |
+---+---+          +---+---+          +---+---+
|   |   |          |   |   |          |   |   |
+---+---+---+      +---+---+---+      +---+---+---+
|   |   |   |      |   |   |   |      |   |   |   |
+---+---+---+---+, +   +---+---+---+, +---+   +---+---+,
|   |   |   |   |  |       |   |   |  |   |       |   |
+---+---+---+---+  +---+---+---+---+  +---+---+---+---+
+---+              +---+              +---+
|   |              |   |              |   |
+---+---+          +---+---+          +   +---+
|   |   |          |   |   |          |       |
+---+---+---+      +---+---+---+      +---+---+---+
|   |   |   |      |   |   |   |      |   |   |   |
+---+---+   +---+, +   +---+   +---+, +---+---+---+---+,
|   |   |       |  |       |       |  |   |   |   |   |
+---+---+---+---+  +---+---+---+---+  +---+---+---+---+
+---+              +---+              +---+
|   |              |   |              |   |
+   +---+          +   +---+          +   +---+
|       |          |       |          |       |
+---+---+---+      +---+---+---+      +---+---+---+
|   |   |   |      |   |   |   |      |   |   |   |
+   +---+---+---+, +---+   +---+---+, +---+---+   +---+,
|       |   |   |  |   |       |   |  |   |   |       |
+---+---+---+---+  +---+---+---+---+  +---+---+---+---+
+---+              +---+              +---+
|   |              |   |              |   |
+   +---+          +---+---+          +---+---+
|       |          |   |   |          |   |   |
+---+---+---+      +   +---+---+      +   +---+---+
|   |   |   |      |       |   |      |       |   |
+   +---+   +---+, +---+---+---+---+, +       +---+---+,
|       |       |  |   |   |   |   |  |           |   |
+---+---+---+---+  +---+---+---+---+  +---+---+---+---+
+---+              +---+              +---+
|   |              |   |              |   |
+---+---+          +---+---+          +---+---+
|   |   |          |   |   |          |   |   |
+   +---+---+      +---+   +---+      +---+   +---+
|       |   |      |   |       |      |   |       |
+---+---+   +---+, +---+---+---+---+, +   +---+---+---+,
|   |   |       |  |   |   |   |   |  |       |   |   |
+---+---+---+---+  +---+---+---+---+  +---+---+---+---+
+---+              +---+              +---+
|   |              |   |              |   |
+---+---+          +   +---+          +   +---+
|   |   |          |       |          |       |
+---+   +---+      +       +---+      +       +---+
|   |       |      |           |      |           |
+---+       +---+, +---+---+---+---+, +           +---+.
|   |           |  |   |   |   |   |  |               |
+---+---+---+---+  +---+---+---+---+  +---+---+---+---+
a(5) = 8+5+5+5+3+8+5+5+5+3+8+3+5+8+5+3+8+2 = 94.

Code Golf Stack Exchange user "Bubbler", Tiling a staircase with staircases.
Rémy Sigrist, PARI program for A335547

Cf. A334617, A335967, A336479.

PARI
```
See Links section.
```

More terms from Rémy Sigrist, Sep 13 2020

cp's OEIS Frontend

A337131 Row lengths of irregular triangle A335967.

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