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

%I A349039 #14 Nov 09 2021 15:02:28
%S A349039 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 A349039 49,31,19,13,13,19,31,49,64,43,28,19,16,19,28,43,64,81,57,39,27,21,21,
%U A349039 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 A349039 Square array T(n, k) read by antidiagonals, n, k >= 0; T(n, k) = n^2 - n*k + k^2.
%C A349039 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 A349039 All terms belong to A003136.
%H A349039 Rémy Sigrist, <a href="/A349039/b349039.txt">Table of n, a(n) for n = 0..10010</a>
%H A349039 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_integer#Euclidean_domain">Eisenstein integers: Euclidean domain</a>
%F A349039 T(n, k) = T(k, n).
%F A349039 T(n, 0) = T(n, n) = n^2.
%F A349039 T(n, k) = A048147(n, k) - A004247(n, k).
%F A349039 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 A349039 Array T(n, k) begins:
%e A349039   n\k|    0   1   2   3   4   5   6   7   8   9   10
%e A349039   ---+----------------------------------------------
%e A349039     0|    0   1   4   9  16  25  36  49  64  81  100
%e A349039     1|    1   1   3   7  13  21  31  43  57  73   91
%e A349039     2|    4   3   4   7  12  19  28  39  52  67   84
%e A349039     3|    9   7   7   9  13  19  27  37  49  63   79
%e A349039     4|   16  13  12  13  16  21  28  37  48  61   76
%e A349039     5|   25  21  19  19  21  25  31  39  49  61   75
%e A349039     6|   36  31  28  27  28  31  36  43  52  63   76
%e A349039     7|   49  43  39  37  37  39  43  49  57  67   79
%e A349039     8|   64  57  52  49  48  49  52  57  64  73   84
%e A349039     9|   81  73  67  63  61  61  63  67  73  81   91
%e A349039    10|  100  91  84  79  76  75  76  79  84  91  100
%t A349039 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 A349039 (PARI) T(n,k) = n^2 - n*k + k^2
%Y A349039 Cf. A003136, A004247, A048147, A073254.
%K A349039 nonn,tabl,easy
%O A349039 0,4
%A A349039 _Rémy Sigrist_, Nov 06 2021