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

0,5

Links

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

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)

Example

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;

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 *)

Prog

(Magma) [(k*(n-k))^2 + 3*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // 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

Crossrefs

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

Sequence in context: A188474 A173046 A173043 * A132787 A181370 A119307

Adjacent sequences: A082043 A082044 A082045 * A082047 A082048 A082049

Keyword

easy,nonn,tabl

Author

Paul Barry, Apr 03 2003

Status

approved