A082110 Array A(n,k) = (k*n)^2 + 5*(k*n) + 1, read by antidiagonals.

1, 1, 1, 1, 7, 1, 1, 15, 15, 1, 1, 25, 37, 25, 1, 1, 37, 67, 67, 37, 1, 1, 51, 105, 127, 105, 51, 1, 1, 67, 151, 205, 205, 151, 67, 1, 1, 85, 205, 301, 337, 301, 205, 85, 1, 1, 105, 267, 415, 501, 501, 415, 267, 105, 1, 1, 127, 337, 547, 697, 751, 697, 547, 337, 127, 1

Offset: 0

Author

Paul Barry, Apr 04 2003

Keywords

• easy
• nonn
• tabl

Examples

			Square array, A(n, k), begins as:
  1,   1,   1,   1,    1,    1,    1,    1,    1, ... A000012;
  1,   7,  15,  25,   37,   51,   67,   85,  105, ... A082111;
  1,  15,  37,  67,  105,  151,  205,  267,  337, ... A082112;
  1,  25,  67, 127,  205,  301,  415,  547,  697, ...
  1,  37, 105, 205,  337,  501,  697,  925, 1185, ...
  1,  51, 151, 301,  501,  751, 1051, 1401, 1801, ...
  1,  67, 205, 415,  697, 1051, 1477, 1975, 2545, ...
  1,  85, 267, 547,  925, 1401, 1975, 2647, 3417, ...
  1, 105, 337, 697, 1185, 1801, 2545, 3417, 4417, ...
Antidiagonals, T(n, k), begins as:
  1;
  1,  1;
  1,  7,   1;
  1, 15,  15,   1;
  1, 25,  37,  25,   1;
  1, 37,  67,  67,  37,   1;
  1, 51, 105, 127, 105,  51,   1;
  1, 67, 151, 205, 205, 151,  67,  1;
  1, 85, 205, 301, 337, 301, 205, 85,  1;
		

Links

• G. C. Greubel, Antidiagonals n = 0..50, flattened

Crossrefs

Cf. A082039, A082043, A082046, A082105, A082111, A082112, A082113, A082114.

Programs

Magma

[(k*(n-k))^2 + 5*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // G. C. Greubel, Dec 22 2022

Mathematica

T[n_, k_]:= (k*(n-k))^2 + 5*(k*(n-k)) + 1;
Table[T[n,k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2022 *)

SageMath

def A082110(n,k): return (k*(n-k))^2 + 5*(k*(n-k)) + 1
flatten([[A082110(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 22 2022

Formula

A(n, k) = (k*n)^2 + 5*(k*n) + 1 (Square array).

A(k, n) = A(n, k).

A(2, k) = A082111(k).

A(3, k) = A082112(k).

A(n, n) = T(2*n, n) = A082113(n) (main diagonal).

T(n, k) = (k*(n-k))^2 + 5*k*(n-k) + 1 (number triangle).

Sum_{k=0..n} T(n, k) = A082114(n) (diagonal sums of the array).

From G. C. Greubel, Dec 22 2022: (Start)

T(n, n-k) = T(n, k).

Sum_{k=0..n} (-1)^k*T(n, k) = (1 - 3*n)*(1 + (-1)^n)/2. (End)