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

0,5

Comments

Arithmetic hexagon of E. Lucas.

References

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

Links

G. C. Greubel, Antidiagonals n = 0..100, flattened

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

Formula

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

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

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

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

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

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

Example

Square array, A(n, k), 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
  ...
Antidiagonals, T(n, k), begin as:
  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;

Mathematica

A216232[n_, k_]:= A216232[n, k]= If[n>k+2 || k>n+4, 0, If[n==0 || k==0, 1, A216232[n-1, k] + A216232[n, k-1]]];
T[n_, k_]:= A216232[k, n-k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 04 2026 *)

Prog

(Magma)
A216232:= function(n, k)
   if n gt k+2 or k gt n+4 then return 0;
   elif n eq 0 or k eq 0 then return 1;
   else return $$(n-1, k) + $$(n, k-1);
   end if;
end function;
T:= func< n, k | A216232(k, n-k) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 04 2026
(SageMath)
@CachedFunction
def A216232(n, k):
    if (n>k+2) or (k>n+4): return 0
    elif (n==0) or (k==0): return 1
    else: return A216232(n-1, k) + A216232(n, k-1)
def T(n, k): return A216232(k, n-k)
print(flatten([[T(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Feb 04 2026

Crossrefs

Cf. A007052, A094803, A094806, A094817, A094821.

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

Sequence in context: A216201 A127514 A078802 * A217765 A237928 A108482

Adjacent sequences: A216229 A216230 A216231 * A216233 A216234 A216235

Keyword

nonn,tabl

Author

Philippe Deléham, Mar 14 2013

Status

approved