A349039 - OEIS

%I #14 Nov 09 2021 15:02:28

%S 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,

%T 49,31,19,13,13,19,31,49,64,43,28,19,16,19,28,43,64,81,57,39,27,21,21,

%U 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

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

%C 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).

%C All terms belong to A003136.

%H Rémy Sigrist, <a href="/A349039/b349039.txt">Table of n, a(n) for n = 0..10010</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_integer#Euclidean_domain">Eisenstein integers: Euclidean domain</a>

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

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

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

%F 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

%e Array T(n, k) begins:

%e   n\k|    0   1   2   3   4   5   6   7   8   9   10

%e   ---+----------------------------------------------

%e     0|    0   1   4   9  16  25  36  49  64  81  100

%e     1|    1   1   3   7  13  21  31  43  57  73   91

%e     2|    4   3   4   7  12  19  28  39  52  67   84

%e     3|    9   7   7   9  13  19  27  37  49  63   79

%e     4|   16  13  12  13  16  21  28  37  48  61   76

%e     5|   25  21  19  19  21  25  31  39  49  61   75

%e     6|   36  31  28  27  28  31  36  43  52  63   76

%e     7|   49  43  39  37  37  39  43  49  57  67   79

%e     8|   64  57  52  49  48  49  52  57  64  73   84

%e     9|   81  73  67  63  61  61  63  67  73  81   91

%e    10|  100  91  84  79  76  75  76  79  84  91  100

%t 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 *)

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

%Y Cf. A003136, A004247, A048147, A073254.

%K nonn,tabl,easy

%O 0,4

%A _Rémy Sigrist_, Nov 06 2021