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

Author

Omar E. Pol, Mar 15 2025

Keywords

• nonn
• more

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).

Examples

			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.