A309559 Triangle read by rows: T(n,k) = 1 + n + k^2/2 - k/2 + k*(n-k), n >= 0, 0 <= k <= n.

1, 2, 2, 3, 4, 4, 4, 6, 7, 7, 5, 8, 10, 11, 11, 6, 10, 13, 15, 16, 16, 7, 12, 16, 19, 21, 22, 22, 8, 14, 19, 23, 26, 28, 29, 29, 9, 16, 22, 27, 31, 34, 36, 37, 37, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 11, 20, 28, 35, 41, 46, 50, 53, 55, 56, 56, 12, 22, 31, 39, 46, 52, 57, 61, 64, 66, 67, 67, 13, 24, 34, 43, 51, 58, 64, 69, 73, 76, 78, 79, 79

Offset: 0

Author

Philip K Hotchkiss, Aug 07 2019

Keywords

• nonn
• tabl

Comments

The rascal triangle (A077028) can be generated by the rule South = (East*West+1)/North or South = East+West+1-North; this number triangle can also be generated by South = East+West+1-North, but there not by an equation of the form South = (East*West+d)/North.

Examples

			For row n=3: T(3,0)=4, T(3,1)=6, T(3,2)=6, T(3,3)=7.
Triangle T begins:
                  1
                2   2
              3   4   4
            4   6   7   7
          5   8  10  11  11
        6  10  13  15  16  16
      7  12  16  19  21  22  22
    8  14  19  23  26  28  29  29
  9  16  22  27  31  34  36  37  37
                 ...
		

Links

• Philip K Hotchkiss, Generalized Rascal Triangles, arXiv:1907.11159 [math.HO], 2019.

Crossrefs

Cf. A077028, A309555, A309557.

Programs

Maple

T := proc(n, k)
   if n<0 or k<0 or k>n then
       0;
   else
       1+n+(1/2)*k^2-(1/2)k +k*(n-k);
   end if;

Mathematica

T[n_,k_]:=1+n+(1/2)*k^2-(1/2)k +k*(n-k); Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten
f[n_] := Table[SeriesCoefficient[(-1+(3-2*x)*y+(-1+x)*y^2)/((-1+x)^2*(-1+y)^3), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n}]; Flatten[Array[f, 13,0]] (* Stefano Spezia, Sep 08 2019 *)

Formula

G.f.: (-1+(3-2*x)*y+(-1+x)*y^2)/((-1+x)^2*(-1+y)^3). - Stefano Spezia, Sep 08 2019