A082046 Square array, A(n, k) = (k*n)^2 + 3*k*n + 1, read by antidiagonals.

1, 1, 1, 1, 5, 1, 1, 11, 11, 1, 1, 19, 29, 19, 1, 1, 29, 55, 55, 29, 1, 1, 41, 89, 109, 89, 41, 1, 1, 55, 131, 181, 181, 131, 55, 1, 1, 71, 181, 271, 305, 271, 181, 71, 1, 1, 89, 239, 379, 461, 461, 379, 239, 89, 1, 1, 109, 305, 505, 649, 701, 649, 505, 305, 109, 1

Offset: 0

Author

Paul Barry, Apr 03 2003

Keywords

• easy
• nonn
• tabl

Examples

			Array, A(n, k), begins as:
  1,  1,   1,   1,   1,    1,    1,    1, ... A000012;
  1,  5,  11,  19,  29,   41,   55,   71, ... A028387;
  1, 11,  29,  55,  89,  131,  181,  239, ... A082108;
  1, 19,  55, 109, 181,  271,  379,  505, ... A069131;
  1, 29,  89, 181, 305,  461,  649,  869, ... ;
  1, 41, 131, 271, 461,  701,  991, 1331, ... ;
  1, 55, 181, 379, 649,  991, 1405, 1891, ... ;
  1, 71, 239, 505, 869, 1331, 1891, 2549, ... ;
Antidiagonals, T(n, k), begin as:
  1;
  1,  1;
  1,  5,   1;
  1, 11,  11,   1;
  1, 19,  29,  19,   1;
  1, 29,  55,  55,  29,   1;
  1, 41,  89, 109,  89,  41,   1;
  1, 55, 131, 181, 181, 131,  55,  1;
  1, 71, 181, 271, 305, 271, 181, 71,  1;
		

Links

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

Crossrefs

Cf. A028387, A057721, A069131, A072025, A082039, A082043, A082047, A082105, A082108.

Programs

Magma

[(k*(n-k))^2 + 3*(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 + 3*(k*(n-k)) + 1;
Table[T[n,k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2022 *)

SageMath

def A082046(n,k): return (k*(n-k))^2 + 3*(k*(n-k)) + 1
flatten([[A082046(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 + 3*k*n + 1 (square array).

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

A(n, n) = T(2*n, n) = A057721(n).

A(n, n+1) = A072025(n).

T(n, k) = (k*(n-k))^2 + 3*k*(n-k) + 1 (antidiagonals).

Sum_{k=0..n} T(n, k) = A082047(n) (antidiagonal sums).

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

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

T(2*n+1, n-1) = T(2*n-1, n-1) = A072025(n-1). (End)