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.

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

Author

Rémy Sigrist, Sep 14 2020

Keywords

• nonn
• base
• tabf
• look

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

• Rémy Sigrist, Table of n, a(n) for n = 1..6766 (rows 1..2^10)
• Rémy Sigrist, Scatterplot of the terms of the first 2^16 rows, flattened
• Rémy Sigrist, PARI program for A335967
• Index entries for sequences related to binary expansion of n

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)).