A380579 Triangle read by rows in which row n lists n successive integers in descending order starting with the n-th positive integer not divisible by 3, with n >= 1 and 1 <= k <= n.

1, 2, 1, 4, 3, 2, 5, 4, 3, 2, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 3, 10, 9, 8, 7, 6, 5, 4, 11, 10, 9, 8, 7, 6, 5, 4, 13, 12, 11, 10, 9, 8, 7, 6, 5, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7

Offset: 1

Author

Omar E. Pol, Jan 29 2025

Keywords

• nonn
• tabl
• easy

Comments

This sequence is mentioned in the Name section of A380580. That sequence represents a template for a Pop-Up pyramid which is related to Combinatorics, Geometry, Number Theory and several tens of integers sequences.

The n-th row of this triangle can be visualized in the template n - 1.

The sum of the n-th row equals the area of the largest polygon in the template n - 1.

In this triangle the last term of the row n is equal to both A237591(n-1,1) and A237593(n-1,1).

The m-th diagonal lists the terms of A008619 but starting from the term whose index is 3*m - 3, with m >= 1.

The column 3*m - 2 lists the terms of A001651 but starting from the m-th term, m >= 1.

The column 3*m - 1 lists the terms of A032766 but starting from the m-th term, m >= 1.

The column 3*m lists the terms of A007494 but starting from the m-th term, m >= 1.

Examples

			Triangle begins:
   1;
   2,  1;
   4,  3,  2;
   5,  4,  3,  2;
   7,  6,  5,  4,  3;
   8,  7,  6,  5,  4,  3;
  10,  9,  8,  7,  6,  5,  4;
  11, 10,  9,  8,  7,  6,  5,  4;
  13, 12, 11, 10,  9,  8,  7,  6,  5;
  14, 13, 12, 11, 10,  9,  8,  7,  6,  5;
  16, 15, 14, 13, 12, 11, 10,  9,  8,  7,  6;
  17, 16, 15, 14, 13, 12, 11, 10,  9,  8,  7,  6;
  19, 18, 17, 16, 15, 14, 13, 12, 11, 10,  9,  8,  7;
  20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10,  9,  8,  7;
  ...
For n = 5 the illustration of the row 5 of the triangle as the column 1 and also as the right border of the 4th slice of A380580 is as shown below:
              _ _ _ _ _ _ _ _ _ _ _ _ _ _
         7   |            _|_            |   7
         6   |          _|_|_|_          |   6
         5   |        _|_ _|_ _|_        |   5
         4   |      _|_ _|_|_|_ _|_      |   4
         3   |_ _ _|_ _ _|_|_|_ _ _|_ _ _|   3
.
The last term of the row 5 is equal to 3, the same as both A237591(4,1) = 3 and A237593(4,1) = 3.
The sum of the 5th row of this triangle is 7 + 6 + 5 + 4 + 3 = 25, the same as the area of largest polygon of the diagram.
.
For n = 6 the illustration of the row 6 of the triangle as the column 1 and also as the right border of the 5th slice of A380580 is as shown below:
            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
       8   |              _|_              |   8
       7   |            _|_|_|_            |   7
       6   |          _|_ _|_ _|_          |   6
       5   |        _|_ _|_|_|_ _|_        |   5
       4   |      _|_ _ _|_|_|_ _ _|_      |   4
       3   |_ _ _|_ _ _|_ _|_ _|_ _ _|_ _ _|   3
.
The last term of the row 6 is equal to 3, the same as both A237591(5,1) = 3 and A237593(5,1) = 3.
The sum of the 6th row of this triangle is 8 + 7 + 6 + 5 + 4 + 3 = 33, the same as the area of the largest polygon of the diagram.
.
For n = 7 the illustration of the row 7 of the triangle as the column 1 and also as the right border of the 6th slice of A380580 is as shown below:
        _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
  10   |                  _|_                  |   10
   9   |                _|_|_|_                |    9
   8   |              _|_ _|_ _|_              |    8
   7   |            _|_ _|_|_|_ _|_            |    7
   6   |          _|_ _ _|_|_|_ _ _|_          |    6
   5   |        _|_ _ _|_ _|_ _|_ _ _|_        |    5
   4   |_ _ _ _|_ _ _ _|_|_|_|_|_ _ _ _|_ _ _ _|    4
.
The last term of the row 7 is equal to 4, the same as both A237591(6,1) = 4 and A237593(6,1) = 4.
The sum of the 7th row of this triangle is 10 + 9 + 8 + 7 + 6 + 5 + 4 = 49, the same as the area of the largest polygon of the diagram.
.
		

Crossrefs

Companion of A380580.

Subsequence of A004736.

Column 1 gives A001651.

Column 2 gives the nonzero terms of A032766.

Column 3 gives the nonzero terms of A007494.

Middle diagonal gives A005408.

Leading diagonal gives A008619.

Cf. A237591, A237593.

Programs

Mathematica

T[n_,k_]:=Floor[(3*n-1)/2]-k+1; Table[T[n,k],{n,13},{k,n}]//Flatten (* Stefano Spezia, Apr 24 2025 *)

Formula

T(n,k) = A001651(n) - k + 1.

G.f.: x*y*(1 + x - x^4*y^2 + x^2*(1 + y) - x^3*y*(1 + 2*y))/((1 - x)^2*(1 + x)*(1 - x*y)^2*(1 + x*y)). - Stefano Spezia, Apr 24 2025