

A155151


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


4



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
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OFFSET

1,1


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^p1 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^p1 = 4*n*k + 2*n + 2*k + 1 = (2*n+1)*(2*k+1) hence 2^p1 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


LINKS

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


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) =


EXAMPLE

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;


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 *)


PROG

(Magma) [4*n*k + 2*n + 2*k + 2: k in [1..n], n in [1..11]]; // 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


CROSSREFS

Cf. A000043, A000668, A016957, A017029, A017341, A054723, A082286, A144650.
Sequence in context: A187397 A152138 A109100 * A104788 A249720 A080360
Adjacent sequences: A155148 A155149 A155150 * A155152 A155153 A155154


KEYWORD

nonn,tabl,easy


AUTHOR

Vincenzo Librandi, Jan 21 2009


EXTENSIONS

Edited by Robert Hochberg, Jun 21 2010


STATUS

approved



