A216232 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 3 or if k-n >= 5, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

1, 1, 1, 1, 2, 1, 1, 3, 3, 0, 1, 4, 6, 3, 0, 0, 5, 10, 9, 0, 0, 0, 5, 15, 19, 9, 0, 0, 0, 0, 20, 34, 28, 0, 0, 0, 0, 0, 20, 54, 62, 28, 0, 0, 0, 0, 0, 0, 74, 116, 90, 0, 0, 0, 0, 0, 0, 0, 74, 190, 206, 90, 0, 0, 0, 0, 0, 0, 0, 0, 264, 396, 296, 0, 0, 0, 0, 0, 0, 0, 0, 0, 264, 660, 692, 296, 0, 0, 0, 0, 0

Offset: 0

Author

Philippe Deléham, Mar 14 2013

Keywords

• nonn
• tabl

Comments

Arithmetic hexagon of E. Lucas.

Examples

			Square array begins:
  1, 1, 1,  1,  1,   0,   0,   0,   0,   0, 0, ... row n=0
  1, 2, 3,  4,  5,   5,   0,   0,   0,   0, 0, ... row n=1
  1, 3, 6, 10, 15,  20,  20,   0,   0,   0, 0, ... row n=2
  0, 3, 9, 19, 34,  54,  74,  74,   0,   0, 0, ... row n=3
  0, 0, 9, 28, 62, 116, 190, 264, 264,   0, 0, ... row n=4
  0, 0, 0, 28, 90, 206, 396, 660, 924, 924, 0, ... row n=5
  ...
Array, read by rows, with 0 omitted:
   1,   1,   1,   1,    1
   1,   2,   3,   4,    5,    5
   1,   3,   6,  10,   15,   20,   20
        3,   9,  19,   34,   54,   74,   74
             9,  28,   62,  116,  190,  264,  264
                 28,   90,  206,  396,  660,  924,  924
                       90,  296,  692, 1352, 2276, 3200, 3200
  ...
		

References

• E. Lucas, Théorie des nombres, Albert Blanchard, Paris, 1958, Tome 1, p. 89.

Links

• E. Lucas, Théorie des nombres, Gauthier-Villars, Paris 1891, Tome 1, p. 89.

Crossrefs

Cf. A007052, A094803, A094806, A094817, A094821

Cf. similar sequences: A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230.

Formula

T(n,n) = A094817(n), for n > 0.

T(n+1,n) = T(n+2,n) = A094803(n).

T(n,n+1) = A007052(n).

T(n,n+2) = A094821(n+1).

T(n,n+3) = T(n,n+4) = A094806(n).

Sum_{k=0..n} T(n-k,k) = A217730(n). - Philippe Deléham, Mar 22 2013