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A309559 Triangle read by rows: T(n,k) = 1 + n + k^2/2 - k/2 + k*(n-k), n >= 0, 0 <= k <= n. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

LINKS

Table of n, a(n) for n=0..90.

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

FORMULA

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

EXAMPLE

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

                 ...

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 *)

CROSSREFS

Cf. A077028, A309555, A309557.

Sequence in context: A070172 A273353 A259197 * A130128 A210556 A208914

Adjacent sequences:  A309556 A309557 A309558 * A309560 A309561 A309562

KEYWORD

nonn,tabl

AUTHOR

Philip K Hotchkiss, Aug 07 2019

STATUS

approved

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Last modified July 13 17:54 EDT 2020. Contains 335689 sequences. (Running on oeis4.)