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

0, 3, 8, 6, 13, 20, 9, 18, 27, 36, 12, 23, 34, 45, 56, 15, 28, 41, 54, 67, 80, 18, 33, 48, 63, 78, 93, 108, 21, 38, 55, 72, 89, 106, 123, 140, 24, 43, 62, 81, 100, 119, 138, 157, 176, 27, 48, 69, 90, 111, 132, 153, 174, 195, 216, 30, 53, 76, 99, 122, 145, 168, 191, 214, 237, 260

Offset: 1

Author

Vincenzo Librandi, Jan 25 2009

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins:
   0;
   3,  8;
   6, 13, 20;
   9, 18, 27, 36;
  12, 23, 34, 45,  56;
  15, 28, 41, 54,  67,  80;
  18, 33, 48, 63,  78,  93, 108;
  21, 38, 55, 72,  89, 106, 123, 140;
  24, 43, 62, 81, 100, 119, 138, 157, 176;
  27, 48, 69, 90, 111, 132, 153, 174, 195, 216;
		

Links

• Vincenzo Librandi, Rows n = 1..100, flattened

Crossrefs

All terms are in A155723.

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

Cf. A162261 (row sums).

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

Diagonals include: A139570, A181510, A271625.

Programs

Magma

/* Triangle: */ [[2*m*n+m+n-4: m in [1..n]]: n in [1..10]]; // Bruno Berselli, Aug 31 2012

Mathematica

Flatten[Table[2 n m + m + n - 4, {n, 10}, {m, n}]] (* Vincenzo Librandi, Mar 01 2012 *)

Python

def A155724(n,k): return 2*n*k+n+k-4
print(flatten([[A155724(n,k) for k in range(1,n+1)] for n in range(1,16)])) # G. C. Greubel, Jan 21 2025

Formula

T(n, k) = A154685(n, k) - 8. - L. Edson Jeffery, Oct 12 2012

2*T(n, k) + 9 = (2*k+1)*(2*n+1). - Vincenzo Librandi, Nov 18 2012

From G. C. Greubel, Jan 21 2025: (Start)

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

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)).

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

Extensions

Edited by N. J. A. Sloane, Jun 23 2010