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A155724 Triangle read by rows: T(n, k) = 2*n*k + n + k - 4.

%I A155724 #33 Jan 23 2025 00:18:22
%S A155724 0,3,8,6,13,20,9,18,27,36,12,23,34,45,56,15,28,41,54,67,80,18,33,48,
%T A155724 63,78,93,108,21,38,55,72,89,106,123,140,24,43,62,81,100,119,138,157,
%U A155724 176,27,48,69,90,111,132,153,174,195,216,30,53,76,99,122,145,168,191,214,237,260
%N A155724 Triangle read by rows: T(n, k) = 2*n*k + n + k - 4.
%H A155724 Vincenzo Librandi, <a href="/A155724/b155724.txt">Rows n = 1..100, flattened</a>
%F A155724 T(n, k) = A154685(n, k) - 8. - _L. Edson Jeffery_, Oct 12 2012
%F A155724 2*T(n, k) + 9 = (2*k+1)*(2*n+1). - _Vincenzo Librandi_, Nov 18 2012
%F A155724 From _G. C. Greubel_, Jan 21 2025: (Start)
%F A155724 T(2*n-1, n) = 4*n^2 + n - 5 (main diagonal).
%F A155724 Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1/4)*( 4*(-1)^(n+1)*n^2 + 2*(2-3*(-1)^n)*n - 7*(1-(-1)^n)).
%F A155724 G.f.: x*y*(3*x + 3*y - 4*x*y)/((1-x)*(1-y))^2. (End)
%e A155724 Triangle begins:
%e A155724    0;
%e A155724    3,  8;
%e A155724    6, 13, 20;
%e A155724    9, 18, 27, 36;
%e A155724   12, 23, 34, 45,  56;
%e A155724   15, 28, 41, 54,  67,  80;
%e A155724   18, 33, 48, 63,  78,  93, 108;
%e A155724   21, 38, 55, 72,  89, 106, 123, 140;
%e A155724   24, 43, 62, 81, 100, 119, 138, 157, 176;
%e A155724   27, 48, 69, 90, 111, 132, 153, 174, 195, 216;
%t A155724 Flatten[Table[2 n m + m + n - 4, {n, 10}, {m, n}]] (* _Vincenzo Librandi_, Mar 01 2012 *)
%o A155724 (Magma) /* Triangle: */ [[2*m*n+m+n-4: m in [1..n]]: n in [1..10]]; // _Bruno Berselli_, Aug 31 2012
%o A155724 (Python)
%o A155724 def A155724(n,k): return 2*n*k+n+k-4
%o A155724 print(flatten([[A155724(n,k) for k in range(1,n+1)] for n in range(1,16)])) # _G. C. Greubel_, Jan 21 2025
%Y A155724 All terms are in A155723.
%Y A155724 Cf. A008585, A016885, A017053, A020705, A154685, A155722.
%Y A155724 Cf. A162261 (row sums).
%Y A155724 Columns k: A008585 (k=1), A016885 (k=2), A017053 (k=3), 9*A020705 (k=4).
%Y A155724 Diagonals include: A139570, A181510, A271625.
%K A155724 nonn,tabl,easy
%O A155724 1,2
%A A155724 _Vincenzo Librandi_, Jan 25 2009
%E A155724 Edited by _N. J. A. Sloane_, Jun 23 2010