A350637 Triangle read by rows: T(n,k) in which row n lists the first n terms of A024916 in reverse order, 1 <= k <= n.

1, 4, 1, 8, 4, 1, 15, 8, 4, 1, 21, 15, 8, 4, 1, 33, 21, 15, 8, 4, 1, 41, 33, 21, 15, 8, 4, 1, 56, 41, 33, 21, 15, 8, 4, 1, 69, 56, 41, 33, 21, 15, 8, 4, 1, 87, 69, 56, 41, 33, 21, 15, 8, 4, 1, 99, 87, 69, 56, 41, 33, 21, 15, 8, 4, 1, 127, 99, 87, 69, 56, 41, 33, 21, 15, 8, 4, 1

Offset: 1

Author

Omar E. Pol, Jan 09 2022

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of cubic cells (or cubes) in the k-th level starting from the base of the stepped pyramid with n levels described in A245092 (see example).

Examples

			Triangle begins:
    1;
    4,  1;
    8,  4,  1;
   15,  8,  4,  1;
   21, 15,  8,  4,  1;
   33, 21, 15,  8,  4,  1;
   41, 33, 21, 15,  8,  4,  1;
   56, 41, 33, 21, 15,  8,  4,  1;
   69, 56, 41, 33, 21, 15,  8,  4,  1;
   87, 69, 56, 41, 33, 21, 15,  8,  4,  1;
   99, 87, 69, 56, 41, 33, 21, 15,  8,  4,  1;
  127, 99, 87, 69, 56, 41, 33, 21, 15,  8,  4,  1;
...
For n = 9 the lateral view and top view of the stepped pyramid described in A245092 look as shown below:
                        _
     9        1        |_|_
     8        4        |_ _|_
     7        8        |_ _|_|_
     6       15        |_ _ _| |_
     5       21        |_ _ _|_ _|_
     4       33        |_ _ _ _| | |_
     3       41        |_ _ _ _|_|_ _|_
     2       56        |_ _ _ _ _|_|_  |_
     1       69        |_ _ _ _ _|_ _|_ _|
.
   Level   Row 9         Lateral view of
     k     T(9,k)      the stepped pyramid
.
                        _ _ _ _ _ _ _ _ _
                       |_| | | | | | | | |
                       |_ _|_| | | | | | |
                       |_ _|  _|_| | | | |
                       |_ _ _|    _|_| | |
                       |_ _ _|  _|  _ _|_|
                       |_ _ _ _|  _| |
                       |_ _ _ _| |_ _|
                       |_ _ _ _ _|
                       |_ _ _ _ _|
.
                           Top view of
                       the stepped pyramid
.
For n = 9 and k = 1 there are 69 cubic cells in the level 1 starting from the base of the stepped pyramid, so T(9,1) = 69.
For n = 9 and k = 9 there is only one cubic cell in the level k = 9 (the top) of the stepped pyramid, so T(9,9) = 1.
The volume of the stepped pyramid (also the total number of cubic cells) represents the 9th term of the convolution of A000203 and A000027 hence it's equal to A175254(9) = 248, equaling the sum of the 9th row of triangle.
		

Links

• Omar E. Pol, Perspective view of the stepped pyramid (first 16 levels)

Crossrefs

Column k gives A024916 starting in row k.

Row sums give A175254.

Cf. A340423 (analog for the tower described in A221529).

Cf. A000027, A000203, A004736, A196020, A235791, A236104, A237591, A237593, A245092, A262626, A272172.

Programs

Mathematica

Join@@Array[Reverse@Array[Sum[#-Mod[#,m],{m,#}]&,#]&,12] (* Giorgos Kalogeropoulos, Jan 12 2022 *)

PARI

row(n) = Vecrev(vector(n, k, sum(i=1, k, k\i*i))); \\ Michel Marcus, Jan 22 2022

Formula

T(n,k) = A024916(A004736(n,k)).

T(n,k) = T(n,k) = A024916(n-k+1).

T(n,k) = Sum_{j=1..n} A272172(j,k).