A144562 Triangle read by rows: T(n, k) = 2*n*k + n + k - 1.

3, 6, 11, 9, 16, 23, 12, 21, 30, 39, 15, 26, 37, 48, 59, 18, 31, 44, 57, 70, 83, 21, 36, 51, 66, 81, 96, 111, 24, 41, 58, 75, 92, 109, 126, 143, 27, 46, 65, 84, 103, 122, 141, 160, 179, 30, 51, 72, 93, 114, 135, 156, 177, 198, 219, 33, 56, 79, 102, 125, 148, 171, 194, 217, 240, 263

Offset: 1

Author

Vincenzo Librandi, Jan 06 2009

Keywords

• nonn
• easy
• tabl

Comments

Rearrangement of A153238, numbers n such that 2*n+3 is not prime (we have 2*T(n,k) + 3 = (2*n+1)*(2*k+1), as 2*n+3 is odd it consists of (at least) two odd factors and all such factors appear by definition).

Examples

			Triangle begins:
   3;
   6, 11;
   9, 16, 23;
  12, 21, 30, 39;
  15, 26, 37, 48,  59;
  18, 31, 44, 57,  70,  83;
  21, 36, 51, 66,  81,  96, 111;
  24, 41, 58, 75,  92, 109, 126, 143;
  27, 46, 65, 84, 103, 122, 141, 160, 179;
  ...
		

Links

• Vincenzo Librandi, Rows n = 1..100, flattened
• Mutsumi Suzuki Vincenzo Librandi's method for sequential primes (Librandi's description in Italian).

Crossrefs

Cf. A067076, A144640, A153238.

Main diagonal gives A142463.

T(2n,n) gives A180863(n+1).

Programs

Magma

[2*n*k+n+k-1: k in [1..n], n in [1..11]]; /* or, see example: */ [[2*n*k+n+k-1: k in [1..n]]: n in [1..9]]; // Bruno Berselli, Dec 04 2011

Maple

A144562:= (n,k) -> 2*n*k +n +k -1; seq(seq(A144562(n,k), k=1..n), n=1..12); # G. C. Greubel, Mar 01 2021

Mathematica

T[n_, k_]:= 2*n*k +n +k -1; Table[T[n, k], {n, 11}, {k, n}]//Flatten

PARI

T(n,k)=2*n*k+n+k-1 \\ Charles R Greathouse IV, Dec 28 2011

Sage

flatten([[2*n*k+n+n-1 for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 01 2021

Formula

Sum_{k=1..n} T(n,k) = n*(2*n^2 + 5*n - 1)/2 = A144640(n). - G. C. Greubel, Mar 01 2021

G.f.: x*y*(3 + 2*x*y + 2*x^3*y^2 - x^2*y*(6 + y))/((1 - x)^2*(1 - x*y)^3). - Stefano Spezia, Nov 04 2024

Extensions

Edited by Ray Chandler, Jan 07 2009