A380573 Number of distinct free polyominoes that are terraces in the first n levels of the stepped pyramid described in A245092.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 28, 29, 31, 32, 33, 34, 35, 36

Offset

1,2

Comments

Consider here that the stepped pyramid is an infinite polycube.

a(n) is also the number of distinct free polyominoes that are parts of the symmetric representations of sigma of the first n positive integers.

Conjecture: the polyomino "I" of width 1 and length k appears for the first time in the level 2*k - 1 starting from the top of the stepped pyramid, k >= 1. In other words: that polyomino appears for the first time in the symmetric representation of sigma(2*k-1).

Links

Table of n, a(n) for n=1..32.

Example

For n = 15 there are 16 distinct free polyominoes that are terraces of the stepped pyramid with 15 levels as shown below, so a(15) = 16.
   _
  |_|  _
     _|_|
    |_|_|      _
       _ _    |_|
      |_|_|  _|_|      _
         _ _|_|_|     |_|
        |_|_|_|       |_|      _
           _ _ _   _ _|_|     |_|
          |_|_|_| |_|_|_|     |_|
             _ _ _|_|_|       |_|
            |_|_|_|_|      _ _|_|              _
               _ _ _ _   _|_|_|_|             |_|
              |_|_|_|_| |_|_|                 |_|
                 _ _ _ _|_|                   |_|
                |_|_|_|_|_|                   |_|
                   _ _ _ _ _   _ _            |_|
                  |_|_|_|_|_| |_|_|      _ _ _|_|
                     _ _ _ _ _|_|      _|_|_|_|_|
                    |_|_|_|_|_|_|    _|_|_|_|
                       _ _ _ _ _ _  |_|_|_|_|
                      |_|_|_|_|_|_| |_|_|_|
                         _ _ _ _ _ _|_|              _
                        |_|_|_|_|_|_|_|    _ _   _ _|_|
                           _ _ _ _ _ _ _  |_|_| |_|_|_|
                          |_|_|_|_|_|_|_| |_|  _|_|_|
                             _ _ _ _ _ _ _|_| |_|_|
                            |_|_|_|_|_|_|_|_|
                               _ _ _ _ _ _ _ _
                              |_|_|_|_|_|_|_|_|
.
Compare with the diagram of A245092.

Crossrefs

Cf. A000105, A000203, A175254, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239931-A239934, A245092, A262626, A347186.

Sequence in context: A321291 A317294 A095736 * A004829 A075592 A285801

Adjacent sequences: A380570 A380571 A380572 * A380574 A380575 A380576

Keyword

nonn,more

Author

Omar E. Pol, Mar 15 2025

Status

approved