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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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A336479 For any number n with k binary digits, a(n) is the number of tilings T of a size k staircase polyomino (as described in A335547) such that the sizes of the polyominoes at the base of T correspond to the lengths of runs of consecutive equal digits in the binary representation of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 5, 3, 2, 3, 1, 1, 1, 2, 8, 5, 11, 18, 8, 5, 3, 5, 11, 7, 3, 5, 1, 1, 1, 2, 13, 8, 26, 42, 18, 11, 26, 42, 94, 58, 29, 47, 13, 8, 5, 8, 29, 18, 36, 58, 26, 16, 7, 11, 26, 16, 5, 8, 1, 1, 1, 2, 21, 13, 60, 97, 42, 26, 87, 141, 317
Offset: 0

Views

Author

Rémy Sigrist, Sep 13 2020

Keywords

Comments

a(0) = 1 corresponds to the empty polyomino.

Examples

			For n = 13, the binary representation of 13 is "1101", so we count the tilings of a size 4 staircase polyomino whose base has the following shape:
      .....
      .   .
      .   .....
      .       .
      +---+   .....
      |   |       .
      |   +---+---+---+
      | 1   1 | 0 | 1 |
      +-------+---+---+
there are 3 such tilings:
      +---+              +---+              +---+
      |   |              |   |              |   |
      +---+---+          +   +---+          +---+---+
      |   |   |          |       |          |   |   |
      +---+---+---+      +---+---+---+      +---+   +---+
      |   |   |   |      |   |   |   |      |   |       |
      |   +---+---+---+  |   +---+---+---+  |   +---+---+---+
      |       |   |   |  |       |   |   |  |       |   |   |
      +-------+---+---+, +-------+---+---+, +-------+---+---+
so a(13) = 3.

Links

Crossrefs

Cf. A000045, A000975, A101211, A123392, A335547.

Programs

  • PARI
    See Links section.

Formula

A335547(n) = Sum_{k = 2^(n-1)..2^n-1} a(k).
a(A000975(n+1)) = A335547(n).
a(2^k-1) = 1 for any k >= 0.
a(2^k) = 1 for any k >= 0.
a(3*2^k) = A000045(k+1) for any k >= 0.
a(7*2^k) = A123392(k) for any k >= 0.

A335967 Irregular table read by rows; if the binary representation of n encodes the last row of a tiling of a staircase polyomino, then the n-th row contains the numbers k whose binary representation encode possible penultimate rows.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 5, 6, 5, 4, 5, 6, 7, 8, 9, 10, 11, 10, 9, 10, 13, 14, 8, 9, 10, 11, 12, 13, 14, 15, 10, 13, 11, 12, 11, 10, 8, 9, 10, 11, 9, 10, 13, 12, 13, 14, 15, 16, 17, 18, 19, 18, 21, 22, 20, 21, 22, 23, 21, 20, 19, 20, 27, 28
Offset: 1

Views

Author

Rémy Sigrist, Sep 14 2020

Keywords

Comments

We consider tilings of staircase polyominoes as described in A335547, and encode rows of such tilings as described in A336479.

Examples

			Triangle begins:
     1: [0]
     2: [1]
     3: [1]
     4: [2]
     5: [2, 3]
     6: [2]
     7: [3]
     8: [4]
     9: [5]
    10: [4, 5, 6, 7]
    11: [5, 6]
    12: [5]
    13: [4, 5]
    ...
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 the 13th row contains 4 and 5 ("100" and "101" in binary).

Links

Crossrefs

Cf. A101211, A335547, A336479, A337131 (row lengths).

Programs

  • PARI
    See Links section.

Formula

A336479(n) = Sum_{k = 1..A337131(n)} A336479(T(n, k)).

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

Views

Author

Peter Kagey, Sep 08 2020

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

Extensions

a(8) from Seiichi Manyama, Sep 09 2020
a(9)-a(10) from Bert Dobbelaere, Sep 12 2020
Showing 1-3 of 3 results.

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