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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.

%I A380579 #75 Jul 23 2025 16:07:13
%S A380579 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,
%T A380579 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,
%U A380579 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
%N 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.
%C A380579 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.
%C A380579 The n-th row of this triangle can be visualized in the template n - 1.
%C A380579 The sum of the n-th row equals the area of the largest polygon in the template n - 1.
%C A380579 In this triangle the last term of the row n is equal to both A237591(n-1,1) and A237593(n-1,1).
%C A380579 The m-th diagonal lists the terms of A008619 but starting from the term whose index is 3*m - 3, with m >= 1.
%C A380579 The column 3*m - 2 lists the terms of A001651 but starting from the m-th term, m >= 1.
%C A380579 The column 3*m - 1 lists the terms of A032766 but starting from the m-th term, m >= 1.
%C A380579 The column 3*m lists the terms of A007494 but starting from the m-th term, m >= 1.
%F A380579 T(n,k) = A001651(n) - k + 1.
%F A380579 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
%e A380579 Triangle begins:
%e A380579    1;
%e A380579    2,  1;
%e A380579    4,  3,  2;
%e A380579    5,  4,  3,  2;
%e A380579    7,  6,  5,  4,  3;
%e A380579    8,  7,  6,  5,  4,  3;
%e A380579   10,  9,  8,  7,  6,  5,  4;
%e A380579   11, 10,  9,  8,  7,  6,  5,  4;
%e A380579   13, 12, 11, 10,  9,  8,  7,  6,  5;
%e A380579   14, 13, 12, 11, 10,  9,  8,  7,  6,  5;
%e A380579   16, 15, 14, 13, 12, 11, 10,  9,  8,  7,  6;
%e A380579   17, 16, 15, 14, 13, 12, 11, 10,  9,  8,  7,  6;
%e A380579   19, 18, 17, 16, 15, 14, 13, 12, 11, 10,  9,  8,  7;
%e A380579   20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10,  9,  8,  7;
%e A380579   ...
%e A380579 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:
%e A380579               _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e A380579          7   |            _|_            |   7
%e A380579          6   |          _|_|_|_          |   6
%e A380579          5   |        _|_ _|_ _|_        |   5
%e A380579          4   |      _|_ _|_|_|_ _|_      |   4
%e A380579          3   |_ _ _|_ _ _|_|_|_ _ _|_ _ _|   3
%e A380579 .
%e A380579 The last term of the row 5 is equal to 3, the same as both A237591(4,1) = 3 and A237593(4,1) = 3.
%e A380579 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.
%e A380579 .
%e A380579 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:
%e A380579             _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e A380579        8   |              _|_              |   8
%e A380579        7   |            _|_|_|_            |   7
%e A380579        6   |          _|_ _|_ _|_          |   6
%e A380579        5   |        _|_ _|_|_|_ _|_        |   5
%e A380579        4   |      _|_ _ _|_|_|_ _ _|_      |   4
%e A380579        3   |_ _ _|_ _ _|_ _|_ _|_ _ _|_ _ _|   3
%e A380579 .
%e A380579 The last term of the row 6 is equal to 3, the same as both A237591(5,1) = 3 and A237593(5,1) = 3.
%e A380579 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.
%e A380579 .
%e A380579 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:
%e A380579         _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e A380579   10   |                  _|_                  |   10
%e A380579    9   |                _|_|_|_                |    9
%e A380579    8   |              _|_ _|_ _|_              |    8
%e A380579    7   |            _|_ _|_|_|_ _|_            |    7
%e A380579    6   |          _|_ _ _|_|_|_ _ _|_          |    6
%e A380579    5   |        _|_ _ _|_ _|_ _|_ _ _|_        |    5
%e A380579    4   |_ _ _ _|_ _ _ _|_|_|_|_|_ _ _ _|_ _ _ _|    4
%e A380579 .
%e A380579 The last term of the row 7 is equal to 4, the same as both A237591(6,1) = 4 and A237593(6,1) = 4.
%e A380579 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.
%e A380579 .
%t A380579 T[n_,k_]:=Floor[(3*n-1)/2]-k+1; Table[T[n,k],{n,13},{k,n}]//Flatten (* _Stefano Spezia_, Apr 24 2025 *)
%Y A380579 Companion of A380580.
%Y A380579 Subsequence of A004736.
%Y A380579 Column 1 gives A001651.
%Y A380579 Column 2 gives the nonzero terms of A032766.
%Y A380579 Column 3 gives the nonzero terms of A007494.
%Y A380579 Middle diagonal gives A005408.
%Y A380579 Leading diagonal gives A008619.
%Y A380579 Cf. A237591, A237593.
%K A380579 nonn,tabl,easy
%O A380579 1,2
%A A380579 _Omar E. Pol_, Jan 29 2025