A144650 Triangle read by rows where T(m,n) = 2m*n + m + n + 1.

5, 8, 13, 11, 18, 25, 14, 23, 32, 41, 17, 28, 39, 50, 61, 20, 33, 46, 59, 72, 85, 23, 38, 53, 68, 83, 98, 113, 26, 43, 60, 77, 94, 111, 128, 145, 29, 48, 67, 86, 105, 124, 143, 162, 181, 32, 53, 74, 95, 116, 137, 158, 179, 200, 221, 35, 58, 81, 104, 127, 150, 173, 196, 219, 242, 265

Offset: 1

Author

Vincenzo Librandi, Jan 13 2009

Keywords

• nonn
• tabl
• easy

Comments

First column: A016789, second column: A016885, third column: A017029, fourth column: A017221, fifth column: A017461. - Vincenzo Librandi, Nov 21 2012

Examples

			Triangle begins:
   5;
   8, 13;
  11, 18, 25;
  14, 23, 32, 41;
  17, 28, 39, 50,  61;
  20, 33, 46, 59,  72,  85;
  23, 38, 53, 68,  83,  98, 113;
  26, 43, 60, 77,  94, 111, 128, 145;
  29, 48, 67, 86, 105, 124, 143, 162, 181;
  32, 53, 74, 95, 116, 137, 158, 179, 200, 221; etc.
		

Links

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

Crossrefs

Cf. A001105, A001844, A003307, A007742, A104275, A142463, A160378.

Columns k: A016789 (k=1), A016885 (k=2), A017029 (k=3), A017221 (k=4), A017461 (k=5).

Programs

Magma

[2*n*k + n + k + 1: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

Mathematica

T[n_,k_]:= 2 n*k + n + k + 1; Table[T[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

SageMath

flatten([[2*n*k+n+k+1 for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 14 2023

Formula

Sum_{n=1..m} T(m, n) = m*(2*m+3)*(m+1)/2 = A160378(n+1) (row sums). - R. J. Mathar, Jan 15 2009, Jan 05 2011

From G. C. Greubel, Oct 14 2023: (Start)

T(n, n) = A001844(n).

T(n, n-1) = A001105(n), n >= 2.

T(n, n-2) = A142463(n-1), n >= 3.

T(n, n-3) = (-1)*A147973(n+2), n >= 4.

Sum_{k=1..n} (-1)^k*T(n, k) = (-1)^n*A007742(floor((n+1)/2)).

G.f.: x*y*(5 - 2*x - 2*x*y - 2*x^2*y + x^2*y^2)/((1-x)^2*(1-x*y)^3). (End)