A155151 Triangle T(n, k) = 4*n*k + 2*n + 2*k + 2, read by rows.

10, 16, 26, 22, 36, 50, 28, 46, 64, 82, 34, 56, 78, 100, 122, 40, 66, 92, 118, 144, 170, 46, 76, 106, 136, 166, 196, 226, 52, 86, 120, 154, 188, 222, 256, 290, 58, 96, 134, 172, 210, 248, 286, 324, 362, 64, 106, 148, 190, 232, 274, 316, 358, 400, 442, 70, 116, 162

Offset: 1

Author

Vincenzo Librandi, Jan 21 2009

Keywords

• nonn
• tabl
• easy

Comments

First column: A016957, second column: A017341, third column: 2*A017029, fourth column: A082286. - Vincenzo Librandi, Nov 21 2012

Conjecture: Let p = prime number. If 2^p belongs to the sequence, then 2^p-1 is not a Mersenne prime. - Vincenzo Librandi, Dec 12 2012

Conjecture is true because if T(n, k) = 2^p with p prime, then 2^p-1 = 4*n*k + 2*n + 2*k + 1 = (2*n+1)*(2*k+1) hence 2^p-1 is not prime. - Michel Marcus, May 31 2015

It appears that T(m,p) = 2^p for Lucasian primes (A002515) greater than 3. For instance: T(44, 11) = 2^11, T(89240, 23) = 2^23. - Michel Marcus, May 28 2015

For n > 1, ascending numbers along the diagonal are also terms of the even principal diagonal of a 2n X 2n spiral (A137928). - Avi Friedlich, May 21 2015

Examples

			Triangle begins
  10;
  16,  26;
  22,  36,  50;
  28,  46,  64,  82;
  34,  56,  78, 100, 122;
  40,  66,  92, 118, 144, 170;
  46,  76, 106, 136, 166, 196, 226;
  52,  86, 120, 154, 188, 222, 256, 290;
  58,  96, 134, 172, 210, 248, 286, 324, 362;
  64, 106, 148, 190, 232, 274, 316, 358, 400, 442;
		

Links

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

Crossrefs

Cf. A000043, A000668, A016957, A017029, A017341, A054723, A082286, A144650.

Programs

Magma

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

Maple

seq(seq( 2*(2*n*k+n+k+1), k=1..n), n=1..15) # G. C. Greubel, Mar 21 2021

Mathematica

T[n_,k_]:=4*n*k + 2*n + 2*k + 2; Table[T[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

Sage

flatten([[2*(2*n*k+n+k+1) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 21 2021

Formula

T(n, k) = 2*A144650(n, k).

Sum_{k=1..n} T(n,k) = n*(2*n^2 + 5*n + 3) = n*A014105(n+2) =

Extensions

Edited by Robert Hochberg, Jun 21 2010