A082105 Array A(n, k) = (k*n)^2 + 4*(k*n) + 1, read by antidiagonals.

1, 1, 1, 1, 6, 1, 1, 13, 13, 1, 1, 22, 33, 22, 1, 1, 33, 61, 61, 33, 1, 1, 46, 97, 118, 97, 46, 1, 1, 61, 141, 193, 193, 141, 61, 1, 1, 78, 193, 286, 321, 286, 193, 78, 1, 1, 97, 253, 397, 481, 481, 397, 253, 97, 1, 1, 118, 321, 526, 673, 726, 673, 526, 321, 118, 1

Offset

0,5

Links

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

Formula

A(n, k) = (k*n)^2 + 4*(k*n) + 1 (square array).

A(n, n) = T(2*n, n) = A082106(n) (main diagonal).

T(n, k) = A(n-k, k) (number triangle).

Sum_{k=0..n} T(n, k) = A082107(n) (diagonal sums).

T(n, n-1) = A028872(n-1), n >= 1.

T(n, n-2) = A082109(n-2), n >= 2.

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

Sum_{k=0..n} (-1)^k * T(n, k) = ((1+(-1)^n)/2)*A016897(n-1).

T(2*n+1, n+1) = A047673(n+1), n >= 0.

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

Example

Array, A(n, k), begins as:
  1,  1,   1,   1,   1,   1, ... A000012;
  1,  6,  13,  22,  33,  46, ... A028872;
  1, 13,  33,  61,  97, 141, ... A082109;
  1, 22,  61, 118, 193, 286, ... ;
  1, 33,  97, 193, 321, 481, ... ;
  1, 46, 141, 286, 481, 726, ... ;
Triangle, T(n, k), begins as:
  1;
  1,  1;
  1,  6,   1;
  1, 13,  13,   1;
  1, 22,  33,  22,   1;
  1, 33,  61,  61,  33,   1;
  1, 46,  97, 118,  97,  46,   1;
  1, 61, 141, 193, 193, 141,  61,  1;
  1, 78, 193, 286, 321, 286, 193, 78,  1;

Mathematica

T[n_, k_]:= (k*(n-k))^2 + 4*(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 + 4*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // G. C. Greubel, Dec 22 2022
(SageMath)
def A082105(n, k): return (k*(n-k))^2 + 4*(k*(n-k)) + 1
flatten([[A082105(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 22 2022

Crossrefs

Cf. A016897, A028872, A047673, A082043, A082046, A082106, A082107, A082109.

Sequence in context: A202673 A202875 A203956 * A353963 A230073 A143210

Adjacent sequences: A082102 A082103 A082104 * A082106 A082107 A082108

Keyword

easy,nonn,tabl

Author

Paul Barry, Apr 03 2003

Status

approved