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

0,5

Links

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

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)

Example

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;

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 *)

Prog

(Magma) [(k*(n-k))^2 + 5*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // 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

Crossrefs

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

Sequence in context: A287326 A131065 A081580 * A275526 A141597 A176561

Adjacent sequences: A082107 A082108 A082109 * A082111 A082112 A082113

Keyword

easy,nonn,tabl

Author

Paul Barry, Apr 04 2003

Status

approved