A309557 Number triangle with T(n,k) = 2 + 3*n - 2*k + 2*k*(n-k) for n >= 0, 0 <= k <= n.

2, 5, 3, 8, 8, 4, 11, 13, 11, 5, 14, 18, 18, 14, 6, 17, 23, 25, 23, 17, 7, 20, 28, 32, 32, 28, 20, 8, 23, 33, 39, 41, 39, 33, 23, 9, 26, 38, 46, 50, 50, 46, 38, 26, 10, 29, 43, 53, 59, 61, 59, 53, 43, 29, 11, 32, 48, 60, 68, 72, 72, 68, 60, 48, 32, 12, 35, 53, 67, 77, 83, 85, 83, 77, 67, 53, 35, 13

Offset

0,1

Comments

The rascal triangle (A077028) can be generated by South = (East*West+1)/North or South = East+West+1-North; this triangle can be generated by South = (East*West+1)/North, South = East+West+2-North.

Links

Table of n, a(n) for n=0..77.

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

Formula

By rows: a(n,k) = 2 + 3*n - 2*k + 2*k*(n-k) n >= 0, 0 <= k <= n.

By antidiagonals: T(r,k) = 2 + 3*k + r + 2*r*k, r,k >= 0.

G.f.: (x*(-1-4*y+y^2)-2*(1-4*y+y^2))/((-1+x)^2*(-1+y)^3). - Stefano Spezia, Sep 08 2019

Example

For row n=3: a(3,0)=11, a(3,1)=13, a(3,2)=11, a(3,3)=5, ...
For antidiagonal r=2: T(2,0)=4, T(2,1)=11, T(2,2)=18, ...
Triangle T begins:
              2
            5   3
          8   8   4
        11  13  11  5
      14  18  18  14  6
    17  23  25  23  17  7
  20  28  32  32  28  20  8
23  33  39  41  39  33  23  9
             ...

Maple

:=proc(n, k)
   if n<0 or k<0 or k>n then
       0;
   else
       2+3*n -2*k +2*k*(n-k);
   end if;

Mathematica

T[n_, k_]:=2+3*n-2*k+2*k*(n-k); Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten
f[n_] := Table[SeriesCoefficient[(x*(-1-4*y+y^2)-2*(1-4*y+y^2))/((-1+x)^2*(-1+y)^3), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n}]; Flatten[Array[f, 12, 0]] (* Stefano Spezia, Sep 08 2019 *)

Crossrefs

Cf. A077028, A309555, A309559.

Sequence in context: A050171 A258406 A136189 * A292965 A302054 A081146

Adjacent sequences: A309554 A309555 A309556 * A309558 A309559 A309560

Keyword

nonn,tabl

Author

Philip K Hotchkiss, Aug 07 2019

Status

approved