A349039 Square array T(n, k) read by antidiagonals, n, k >= 0; T(n, k) = n^2 - n*k + k^2.

0, 1, 1, 4, 1, 4, 9, 3, 3, 9, 16, 7, 4, 7, 16, 25, 13, 7, 7, 13, 25, 36, 21, 12, 9, 12, 21, 36, 49, 31, 19, 13, 13, 19, 31, 49, 64, 43, 28, 19, 16, 19, 28, 43, 64, 81, 57, 39, 27, 21, 21, 27, 39, 57, 81, 100, 73, 52, 37, 28, 25, 28, 37, 52, 73, 100, 121, 91, 67, 49, 37, 31, 31, 37, 49, 67, 91, 121

Offset

0,4

Comments

T(n, k) is the norm of the Eisenstein integer n + k*w (where w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity).

All terms belong to A003136.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10010

Wikipedia, Eisenstein integers: Euclidean domain

Formula

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

T(n, 0) = T(n, n) = n^2.

T(n, k) = A048147(n, k) - A004247(n, k).

G.f.: (x - 5*x*y + y*(1 + y) + x^2*(1 + y^2))/((1 - x)^3*(1 - y)^3). - Stefano Spezia, Nov 08 2021

Example

Array T(n, k) begins:
  n\k|    0   1   2   3   4   5   6   7   8   9   10
  ---+----------------------------------------------
    0|    0   1   4   9  16  25  36  49  64  81  100
    1|    1   1   3   7  13  21  31  43  57  73   91
    2|    4   3   4   7  12  19  28  39  52  67   84
    3|    9   7   7   9  13  19  27  37  49  63   79
    4|   16  13  12  13  16  21  28  37  48  61   76
    5|   25  21  19  19  21  25  31  39  49  61   75
    6|   36  31  28  27  28  31  36  43  52  63   76
    7|   49  43  39  37  37  39  43  49  57  67   79
    8|   64  57  52  49  48  49  52  57  64  73   84
    9|   81  73  67  63  61  61  63  67  73  81   91
   10|  100  91  84  79  76  75  76  79  84  91  100

Mathematica

T[n_, k_] := n^2 - n*k + k^2; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 08 2021 *)

Prog

(PARI) T(n, k) = n^2 - n*k + k^2

Crossrefs

Cf. A003136, A004247, A048147, A073254.

Sequence in context: A133819 A344686 A274092 * A021245 A007891 A165487

Adjacent sequences: A349036 A349037 A349038 * A349040 A349041 A349042

Keyword

nonn,tabl,easy

Author

Rémy Sigrist, Nov 06 2021

Status

approved