A341148 Triangle read by rows: T(n,k) is number of cubes in the k-th vertical slice of the polycube called "tower" described in A221529 where n is the longest side of its base, 1 <= k <= n.

1, 2, 2, 4, 3, 2, 7, 6, 4, 3, 12, 10, 7, 3, 3, 19, 17, 12, 9, 5, 4, 30, 26, 20, 13, 8, 4, 4, 45, 41, 31, 23, 16, 10, 5, 5, 67, 60, 48, 34, 25, 15, 11, 5, 5, 97, 89, 71, 55, 39, 28, 17, 12, 6, 6, 139, 127, 104, 78, 60, 40, 28, 17, 11, 6, 6, 195, 181, 149, 118, 89, 65, 45, 32, 21, 15, 7, 7

Offset: 1

Author

Omar E. Pol, Feb 06 2021

Keywords

• nonn
• tabl

Comments

The row sums of triangle give A066186 because the correspondence divisor/part. For more information see A338156.

For further information about the tower see A221529.

Examples

			Triangle begins:
    1;
    2,   2;
    4,   3,   2;
    7,   6,   4,   3;
   12,  10,   7,   3,  3;
   19,  17,  12,   9,  5,  4;
   30,  26,  20,  13,  8,  4,  4;
   45,  41,  31,  23, 16, 10,  5,  5;
   67,  60,  48,  34, 25, 15, 11,  5,  5;
   97,  89,  71,  55, 39, 28, 17, 12,  6,  6;
  139, 127, 104,  78, 60, 40, 28, 17, 11,  6,  6;
  195, 181, 149, 118, 89, 65, 45, 32, 21, 15,  7,  7;
...
Illustration of initial terms:
              Top view
  n   k       of the tower       Heights        T(n,k)
               _
  1   1       |_|                1                 1
.              _ _
  2   1       |   |              1 1               2
  2   2       |_ _|              1 1               2
.              _ _ _
  3   1       |_|   |            2 1 1             4
  3   2       |    _|            1 1 1             3
  3   3       |_ _|              1 1               2
.              _ _ _ _
  4   1       |_| |   |          3 2 1 1           7
  4   2       |_ _|   |          2 2 1 1           6
  4   3       |      _|          1 1 1 1           4
  4   4       |_ _ _|            1 1 1             3
.              _ _ _ _ _
  5   1       |_| | |   |        5 3 2 1 1        12
  5   2       |_ _|_|   |        3 3 2 1 1        10
  5   3       |_ _|  _ _|        2 2 1 1 1         7
  5   4       |     |            1 1 1             3
  5   5       |_ _ _|            1 1 1             3
.              _ _ _ _ _ _
  6   1       |_| | | |   |      7 5 3 2 1 1      19
  6   2       |_ _|_| |   |      5 5 3 2 1 1      17
  6   3       |_ _|  _|   |      3 3 2 2 1 1      12
  6   4       |_ _ _|    _|      2 2 2 1 1 1       9
  6   5       |        _|        1 1 1 1 1         5
  6   6       |_ _ _ _|          1 1 1 1           4
.
The levels of the terraces of the tower are the partition numbers A000041 starting from the base.
Note that the top view of the tower is essentially the same as the top view of the stepped pyramid described in A245092 except that in the tower both the symmetric representation of sigma(n) and the symmetric representation of sigma(n-1) are unified in the level 1 of the structure because the first two partitions numbers A000041 are [1, 1].
		

Crossrefs

Column 1 gives A000070.

Leading diagonal gives A080513.

Row sums give A066186.

Cf. A000041, A024916, A236104, A237270, A237271, A237593, A221529, A245092, A262626, A336811, A336812, A338156, A340035.