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

0,5

Links

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

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

Example

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;

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 *)

Prog

(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
(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

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.

Sequence in context: A199065 A152237 A176282 * A177944 A174006 A124216

Adjacent sequences: A082040 A082041 A082042 * A082044 A082045 A082046

Keyword

easy,nonn,tabl

Author

Paul Barry, Apr 03 2003

Status

approved