A083487 Triangle read by rows: T(n,k) = 2*n*k + n + k (1 <= k <= n).

4, 7, 12, 10, 17, 24, 13, 22, 31, 40, 16, 27, 38, 49, 60, 19, 32, 45, 58, 71, 84, 22, 37, 52, 67, 82, 97, 112, 25, 42, 59, 76, 93, 110, 127, 144, 28, 47, 66, 85, 104, 123, 142, 161, 180, 31, 52, 73, 94, 115, 136, 157, 178, 199, 220, 34, 57, 80, 103, 126, 149, 172, 195, 218, 241, 264

Offset: 1

Author

Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), Jun 09 2003

Keywords

• nonn
• tabl

Comments

T(n,k) gives number of edges (of unit length) in a k X n grid.

The values 2*T(n,k)+1 = (2*n+1)*(2*k+1) are nonprime and therefore in A047845.

Examples

			Triangle begins:
   4;
   7, 12;
  10, 17, 24;
  13, 22, 31, 40;
  16, 27, 38, 49,  60;
  19, 32, 45, 58,  71,  84;
  22, 37, 52, 67,  82,  97, 112;
  25, 42, 59, 76,  93, 110, 127, 144;
  28, 47, 66, 85, 104, 123, 142, 161, 180;
		

Links

• Vincenzo Librandi, Rows n = 1..100, flattened
• Alain Kraus, Cours-Arithmétique et algèbre, 2016-2017, Université de Paris VI. See Exercice 6 p. 13.
• OEIS Wiki, Odd composites

Crossrefs

Cf. A000217, A016777, A016873, A017017, A017209, A017449, A033954.

Cf. A056220, A082652, A083003, A090288, A182868, A186113, A271625.

See A151890 for another version.

Programs

Magma

[(2*n*k + n + k): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Jun 01 2014

Mathematica

T[n_,k_]:= 2 n k + n + k; Table[T[n, k], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Jun 01 2014 *)

Python

def T(r, c): return 2*r*c + r + c
a = [T(r, c) for r in range(12) for c in range(1, r+1)]
print(a) # Michael S. Branicky, Sep 07 2022

SageMath

flatten([[2*n*k +n +k for k in range(1,n+1)] for n in range(1,14)]) # G. C. Greubel, Oct 17 2023

Formula

From G. C. Greubel, Oct 17 2023: (Start)

T(n, 1) = A016777(n).

T(n, 2) = A016873(n).

T(n, 3) = A017017(n).

T(n, 4) = A017209(n).

T(n, 5) = A017449(n).

T(n, 6) = A186113(n).

T(n, n-1) = A056220(n).

T(n, n-2) = A090288(n-2).

T(n, n-3) = A271625(n-2).

T(n, n) = 4*A000217(n).

T(2*n, n) = A033954(n).

Sum_{k=1..n} T(n, k) = A162254(n).

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A182868((n+1)/2) if n is odd otherwise A182868(n/2) + 1. (End)

Extensions

Edited by N. J. A. Sloane, Jul 23 2009

Name edited by Michael S. Branicky, Sep 07 2022