A319840 Table read by antidiagonals: T(n, k) is the number of elements on the perimeter of an n X k matrix.

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 8, 8, 5, 6, 10, 10, 10, 10, 6, 7, 12, 12, 12, 12, 12, 7, 8, 14, 14, 14, 14, 14, 14, 8, 9, 16, 16, 16, 16, 16, 16, 16, 9, 10, 18, 18, 18, 18, 18, 18, 18, 18, 10, 11, 20, 20, 20, 20, 20, 20, 20, 20, 20, 11, 12, 22, 22, 22

Offset: 1

Author

Stefano Spezia, Sep 29 2018

Keywords

• nonn
• tabl

Comments

The table T(n, k) can be indifferently read by ascending or descending antidiagonals.

Examples

			The table T starts in row n=1 with columns k >= 1 as:
   1   2   3   4   5   6   7   8   9  10 ...
   2   4   6   8  10  12  14  16  18  20 ...
   3   6   8  10  12  14  16  18  20  22 ...
   4   8  10  12  14  16  18  20  22  24 ...
   5  10  12  14  16  18  20  22  24  26 ...
   6  12  14  16  18  20  22  24  26  28 ...
   7  14  16  18  20  22  24  26  28  30 ...
   8  16  18  20  22  24  26  28  30  32 ...
   9  18  20  22  24  26  28  30  32  34 ...
  10  20  22  24  26  28  30  32  34  36 ...
  ...
The triangle X(n, k) begins
  n\k|   1   2   3   4   5   6   7   8   9  10
  ---+----------------------------------------
   1 |   1
   2 |   2   2
   3 |   3   4   3
   4 |   4   6   6   4
   5 |   5   8   8   8   5
   6 |   6  10  10  10  10   6
   7 |   7  12  12  12  12  12   7
   8 |   8  14  14  14  14  14  14   8
   9 |   9  16  16  16  16  16  16  16   9
  10 |  10  18  18  18  18  18  18  18  18  10
  ...
		

Crossrefs

Cf. A000027 (1st column/right diagonal of the triangle or 1st row/column of the table), A005843 (2nd row/column of the table, or 2nd column of the triangle), A008574 (main diagonal of the table), A005893 (row sum of the triangle).

Cf. A003991 (the number of elements in an n X k matrix).

Cf. A131821, A318274, A154325.

Programs

Magma

[[k lt 3 or n+1-k lt 3 select (n+1-k)*k else 2*n-2: k in [1..n]]: n in [1..10]]; // triangle output

Maple

a := (n, k) -> (n+1-k)*k-(n-1-k)*(k-2)*(limit(Heaviside(min(n+1-k, k)-3+x), x = 0, right)): seq(seq(a(n, k), k = 1 .. n), n = 1 .. 20)

Mathematica

Flatten[Table[(n + 1 - k) k-(n-1-k)*(k-2)Limit[HeavisideTheta[Min[n+1-k,k]-3+x], x->0, Direction->"FromAbove"  ],{n, 20}, {k, n}]] (* or *)
f[n_] := Table[SeriesCoefficient[(x y - x^3 y^3)/((-1 + x)^2 (-1 + y)^2), {x, 0, i + 1 - j}, {y, 0, j}], {i, n, n}, {j, 1, n}]; Flatten[Array[f,20]]

PARI

T(n, k) = if ((n+1-k<3) || (k<3), (n+1-k)*k, 2*n-2);
tabl(nn) = for(i=1, nn, for(j=1, i, print1(T(i, j), ", ")); print);
tabl(20) \\ triangle output

Formula

T(n, k) = n*k - (n - 2)*(k - 2)*H(min(n, k) - 3), where H(x) is the Heaviside step function, taking H(0) = 1.

G.f. as rectangular array: (x*y - x^3*y^3)/((-1 + x)^2*(-1 + y)^2).

X(n, k) = A131821(n, k)*A318274(n - 1, k)*A154325(n - 1, k). - Franck Maminirina Ramaharo, Nov 18 2018