A082043 Square array, A(n, k) = (k*n)^2 + 2*k*n + 1, read by antidiagonals.

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 25, 16, 1, 1, 25, 49, 49, 25, 1, 1, 36, 81, 100, 81, 36, 1, 1, 49, 121, 169, 169, 121, 49, 1, 1, 64, 169, 256, 289, 256, 169, 64, 1, 1, 81, 225, 361, 441, 441, 361, 225, 81, 1, 1, 100, 289, 484, 625, 676, 625, 484, 289, 100, 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,     1, ... A000012;
  1,   4,   9,   16,   25,   36,   49,   64,    81, ... A000290;
  1,   9,  25,   49,   81,  121,  169,  225,   289, ... A016754;
  1,  16,  49,  100,  169,  256,  361,  484,   625, ... A016778;
  1,  25,  81,  169,  289,  441,  625,  841,  1089, ... A016814;
  1,  36, 121,  256,  441,  676,  961, 1296,  1681, ... A016862;
  1,  49, 169,  361,  625,  961, 1369, 1849,  2401, ... A016922;
  1,  64, 225,  484,  841, 1296, 1849, 2500,  3249, ... A016994;
  1,  81, 289,  625, 1089, 1681, 2401, 3249,  4225, ... A017078;
  1, 100, 361,  784, 1369, 2116, 3025, 4096,  5329, ... A017174;
  1, 121, 441,  961, 1681, 2601, 3721, 5041,  6561, ... A017282;
  1, 144, 529, 1156, 2025, 3136, 4489, 6084,  7921, ... A017402;
  1, 169, 625, 1369, 2401, 3721, 5329, 7225,  9409, ... A017534;
  1, 196, 729, 1600, 2809, 4356, 6241, 8464, 11025, ... ;
Antidiagonals, T(n, k), begin as:
  1;
  1,   1;
  1,   4,   1;
  1,   9,   9,   1;
  1,  16,  25,  16,   1;
  1,  25,  49,  49,  25,   1;
  1,  36,  81, 100,  81,  36,   1;
  1,  49, 121, 169, 169, 121,  49,   1;
  1,  64, 169, 256, 289, 256, 169,  64,   1;
  1,  81, 225, 361, 441, 441, 361, 225,  81,   1;
  1, 100, 289, 484, 625, 676, 625, 484, 289, 100,  1;
		

Links

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

Crossrefs

Rows include A000290, A016754, A016778, A016814, A016862, A016922, A016994, A017078, A017174, A017282, A017402, A017534.

Diagonals include A000583, A058031, A062938, A082044 (main diagonal).

Diagonal sums (row sums if viewed as number triangle) are A082045.

Cf. A082039, A082045, A082046, A082105.

Programs

Magma

A082043:= func< n,k | (k*(n-k))^2 +2*k*(n-k) +1 >;
[A082043(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 24 2022

Mathematica

T[n_, k_]:= (k*(n-k))^2 +2*k*(n-k) +1;
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 24 2022 *)

SageMath

def A082043(n,k): return (k*(n-k))^2 +2*k*(n-k) +1
flatten([[A082043(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Dec 24 2022

Formula

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

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

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

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

A(n, n) = T(2*n, n) = A082044(n).

A(n, n-1) = T(2*n+1, n-1) = A058031(n), n >= 1.

A(n, n-2) = T(2*(n-1), n) = A000583(n-1), n >= 2.

A(n, n-3) = T(2*n-3, n) = A062938(n-3), n >= 3.

Sum_{k=0..n} T(n, k) = A082045(n) (diagonal sums).

Sum_{k=0..n} (-1)^k * T(n, k) = (1/4)*(1+(-1)^n)*(2 - 3*n). - G. C. Greubel, Dec 24 2022