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A132111 Triangle read by rows: T(n,k) = n^2 + k*n + k^2, 0 <= k <= n.

%I A132111 #20 Jun 14 2021 04:15:24
%S A132111 0,1,3,4,7,12,9,13,19,27,16,21,28,37,48,25,31,39,49,61,75,36,43,52,63,
%T A132111 76,91,108,49,57,67,79,93,109,127,147,64,73,84,97,112,129,148,169,192,
%U A132111 81,91,103,117,133,151,171,193,217,243,100,111,124,139,156,175,196,219
%N A132111 Triangle read by rows: T(n,k) = n^2 + k*n + k^2, 0 <= k <= n.
%C A132111 Permutation of A003136, the Loeschian numbers. [This is false - some terms are repeated, the first being 49. - _Joerg Arndt_, Dec 18 2015]
%C A132111 Row sums give A132112.
%C A132111 Central terms give A033582.
%C A132111 T(n,k+1) = T(n,k) + n + 2*k + 1;
%C A132111 T(n+1,k) = T(n,k) + 2*n + k + 1;
%C A132111 T(n+1,k+1) = T(n,k) + 3*(n+k+1);
%C A132111 T(n,0) = A000290(n);
%C A132111 T(n,1) = A002061(n+1) for n>0;
%C A132111 T(n,2) = A117950(n+1) for n>1;
%C A132111 T(n,n-2) = A056107(n-1) for n>1;
%C A132111 T(n,n-1) = A003215(n-1) for n>0;
%C A132111 T(n,n) = A033428(n).
%C A132111 T(n,k) is the norm N(alpha) of the integer alpha = n*1 - k*omega, where omega = exp(2*Pi*i/3) = (-1 + i*sqrt(3))/2 in the imaginary quadratic number field Q(sqrt(-3)): N = |alpha|^2 = (n + k/2)^2 + (3/4)*k^2  = n^2 + n*k + k^2 = T(n,k), with n >= 0, and k <= n. See also triangle A073254 for T(n,-k). - _Wolfdieter Lang_, Jun 13 2021
%e A132111 From _Philippe Deléham_, Apr 16 2014: (Start)
%e A132111 Triangle begins:
%e A132111    0;
%e A132111    1,  3;
%e A132111    4,  7,  12;
%e A132111    9, 13,  19,  27;
%e A132111   16, 21,  28,  37,  48;
%e A132111   25, 31,  39,  49,  61,  75;
%e A132111   36, 43,  52,  63,  76,  91, 108;
%e A132111   49, 57,  67,  79,  93, 109, 127, 147;
%e A132111   64, 73,  84,  97, 112, 129, 148, 169, 192;
%e A132111   81, 91, 103, 117, 133, 151, 171, 193, 217, 243;
%e A132111   ...
%e A132111 (End)
%t A132111 Flatten[Table[n^2+k*n+k^2,{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Jun 10 2013 *)
%Y A132111 Cf. A073254.
%K A132111 nonn,tabl,easy
%O A132111 0,3
%A A132111 _Reinhard Zumkeller_, Aug 10 2007