A154685 Triangle read by rows: T(n, k) = 2*n*k + n + k + 4.

8, 11, 16, 14, 21, 28, 17, 26, 35, 44, 20, 31, 42, 53, 64, 23, 36, 49, 62, 75, 88, 26, 41, 56, 71, 86, 101, 116, 29, 46, 63, 80, 97, 114, 131, 148, 32, 51, 70, 89, 108, 127, 146, 165, 184, 35, 56, 77, 98, 119, 140, 161, 182, 203, 224, 38, 61, 84, 107, 130, 153, 176, 199, 222, 245, 268

Offset: 1

Author

Vincenzo Librandi, Jan 18 2009

Keywords

• nonn
• tabl
• easy

Comments

The terms form a subset of A153039 because 2*T(n, k) - 7 = (2*n+1)*(2*k+1) are not prime.

Examples

			Triangle begins:
   8;
  11, 16;
  14, 21, 28;
  17, 26, 35, 44;
  20, 31, 42, 53,  64;
  23, 36, 49, 62,  75,  88;
  26, 41, 56, 71,  86, 101, 116;
  29, 46, 63, 80,  97, 114, 131, 148;
  32, 51, 70, 89, 108, 127, 146, 165, 184;
  35, 56, 77, 98, 119, 140, 161, 182, 203, 224;
		

Links

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

Crossrefs

Cf. A089192, A153039.

Cf. A151675 (row sums).

Similar triangle: A155724.

Columns k: A016789 (k=1), A016861 (k=2).

Main diagonal: A137882, A271649.

Programs

Magma

A154685:= func< n,k | 2*n*k+n+k+4 >;
[A154685(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jan 21 2025

Mathematica

Flatten@Table[2*n*m+m+n+4,{n,20},{m,n}] (* Vincenzo Librandi, Jan 29 2012 *)

PARI

for(m=1,9,for(n=1,m,print1(2*m*n+m+n+4", "))) \\ Charles R Greathouse IV, Dec 27 2011

Python

def A154685(n,k): return 2*n*k+n+k+4
print(flatten([[A154685(n,k) for k in range(1,n+1)] for n in range(1,16)])) # G. C. Greubel, Jan 21 2025

Formula

Sum_{k=1..n} T(n, k) = A151675(n). - N. J. A. Sloane, May 31 2009

T(n, k) = A155724(n,k) + 8. - L. Edson Jeffery, Oct 12 2012

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

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

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1/4)*(9*(1-(-1)^n) + 2*(2-3*(-1)^n)*n - 4*(-1)^n*n^2).

G.f.: x*y*(8 - 5*(x+y) + 4*x*y)/((1-x)*(1-y))^2.

E.g.f.: 4 - (4+x)*exp(x) - (4+y)*exp(y) + (4+x+y+2*x*y)*exp(x+y).

(End)

Extensions

Clarified comment. - R. J. Mathar, Jan 24 2009