A027692 a(n) = n^2 + n + 7.

7, 9, 13, 19, 27, 37, 49, 63, 79, 97, 117, 139, 163, 189, 217, 247, 279, 313, 349, 387, 427, 469, 513, 559, 607, 657, 709, 763, 819, 877, 937, 999, 1063, 1129, 1197, 1267, 1339, 1413, 1489, 1567, 1647, 1729, 1813, 1899, 1987, 2077, 2169, 2263, 2359, 2457, 2557

Offset: 0

Author

Patrick De Geest

Keywords

• nonn
• easy

Comments

Integers k for which the discriminant of x^3 - k*x - k is a square. - Jacob A. Siehler, Mar 14 2009

Integers k for which the Galois group of the polynomial x^3 - k*x - k over Q is a cyclic group of order 3. See Conrad, Corollary 2.5. - Peter Bala, Oct 17 2021

From Peter Bala, Nov 18 2021: (Start)

Integers k such that 4*k - 27 is a square.

Integers k for which the Galois group of the polynomial x^3 + k*(x + 1)^2 over Q is the cyclic group C_3 (apply Conrad, Corollary 2.5 and Uchida, Lemma 1).

For the primes in this list see A005471. (End)

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..3000
• Keith Conrad, Galois groups of cubics and quartics (not in characteristic 2).
• Patrick De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X).
• Koji Uchida, Class numbers of cubic cyclic fields, Journal of the Mathematical Society of Japan, Vol. 26, No. 3, 1974, pp. 447-453.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A002522, A005471 (subset of primes), A109007, A176271.

Programs

GAP

List([0..50],n->n^2+n+7); # Muniru A Asiru, Jul 15 2018

Maple

A027692 := proc(n)
    n*(n+1)+7 ;
end proc: # R. J. Mathar, Jun 06 2019

Mathematica

f[n_]:=n^2+n+7;f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011 *)

PARI

a(n)=n^2+n+7 \\ Charles R Greathouse IV, Jun 11 2015

Formula

For n > 2: a(n) = A176271(n+1,4). - Reinhard Zumkeller, Apr 13 2010

a(n) = 2*n + a(n-1) (with a(0)=7). - Vincenzo Librandi, Aug 05 2010

G.f.: (-7 + 12*x - 7*x^2)/(x-1)^3. - R. J. Mathar, Feb 06 2011

a(n+1) = n^2 + 3*n + 9, see A005471. - R. J. Mathar, Jun 06 2019

a(n) mod 6 = A109007(n+2). - R. J. Mathar, Jun 06 2019

Sum_{n>=0} 1/a(n) = Pi*tanh(Pi*3*sqrt(3)/2)/(3*sqrt(3)). - Amiram Eldar, Jan 17 2021

From Elmo R. Oliveira, Oct 30 2024: (Start)

E.g.f.: exp(x)*(7 + 2*x + x^2).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)