A155724 - OEIS

%I #33 Jan 23 2025 00:18:22

%S 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 63,78,93,108,21,38,55,72,89,106,123,140,24,43,62,81,100,119,138,157,

%U 176,27,48,69,90,111,132,153,174,195,216,30,53,76,99,122,145,168,191,214,237,260

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

%H Vincenzo Librandi, <a href="/A155724/b155724.txt">Rows n = 1..100, flattened</a>

%F T(n, k) = A154685(n, k) - 8. - _L. Edson Jeffery_, Oct 12 2012

%F 2*T(n, k) + 9 = (2*k+1)*(2*n+1). - _Vincenzo Librandi_, Nov 18 2012

%F From _G. C. Greubel_, Jan 21 2025: (Start)

%F T(2*n-1, n) = 4*n^2 + n - 5 (main diagonal).

%F 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 G.f.: x*y*(3*x + 3*y - 4*x*y)/((1-x)*(1-y))^2. (End)

%e Triangle begins:

%e    0;

%e    3,  8;

%e    6, 13, 20;

%e    9, 18, 27, 36;

%e   12, 23, 34, 45,  56;

%e   15, 28, 41, 54,  67,  80;

%e   18, 33, 48, 63,  78,  93, 108;

%e   21, 38, 55, 72,  89, 106, 123, 140;

%e   24, 43, 62, 81, 100, 119, 138, 157, 176;

%e   27, 48, 69, 90, 111, 132, 153, 174, 195, 216;

%t Flatten[Table[2 n m + m + n - 4, {n, 10}, {m, n}]] (* _Vincenzo Librandi_, Mar 01 2012 *)

%o (Magma) /* Triangle: */ [[2*m*n+m+n-4: m in [1..n]]: n in [1..10]]; // _Bruno Berselli_, Aug 31 2012

%o (Python)

%o def A155724(n,k): return 2*n*k+n+k-4

%o 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 All terms are in A155723.

%Y Cf. A008585, A016885, A017053, A020705, A154685, A155722.

%Y Cf. A162261 (row sums).

%Y Columns k: A008585 (k=1), A016885 (k=2), A017053 (k=3), 9*A020705 (k=4).

%Y Diagonals include: A139570, A181510, A271625.

%K nonn,tabl,easy

%O 1,2

%A _Vincenzo Librandi_, Jan 25 2009

%E Edited by _N. J. A. Sloane_, Jun 23 2010