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

%I A027692 #64 Oct 30 2024 21:39:49
%S A027692 7,9,13,19,27,37,49,63,79,97,117,139,163,189,217,247,279,313,349,387,
%T A027692 427,469,513,559,607,657,709,763,819,877,937,999,1063,1129,1197,1267,
%U A027692 1339,1413,1489,1567,1647,1729,1813,1899,1987,2077,2169,2263,2359,2457,2557
%N A027692 a(n) = n^2 + n + 7.
%C A027692 Integers k for which the discriminant of x^3 - k*x - k is a square. - _Jacob A. Siehler_, Mar 14 2009
%C A027692 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
%C A027692 From _Peter Bala_, Nov 18 2021: (Start)
%C A027692 Integers k such that 4*k - 27 is a square.
%C A027692 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).
%C A027692 For the primes in this list see A005471. (End)
%H A027692 Muniru A Asiru, <a href="/A027692/b027692.txt">Table of n, a(n) for n = 0..3000</a>
%H A027692 Keith Conrad, <a href="http://www.ms.uky.edu/~sohum/ma561/notes/workspace/books/cubicquartic.pdf">Galois groups of cubics and quartics (not in characteristic 2)</a>.
%H A027692 Patrick De Geest, <a href="http://www.worldofnumbers.com/quasimor.htm">Palindromic Quasi_Over_Squares of the form n^2+(n+X)</a>.
%H A027692 Koji Uchida, <a href="https://doi.org/10.2969/jmsj/02630447">Class numbers of cubic cyclic fields</a>, Journal of the Mathematical Society of Japan, Vol. 26, No. 3, 1974, pp. 447-453.
%H A027692 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A027692 For n > 2: a(n) = A176271(n+1,4). - _Reinhard Zumkeller_, Apr 13 2010
%F A027692 a(n) = 2*n + a(n-1) (with a(0)=7). - _Vincenzo Librandi_, Aug 05 2010
%F A027692 G.f.: (-7 + 12*x - 7*x^2)/(x-1)^3. - _R. J. Mathar_, Feb 06 2011
%F A027692 a(n+1) = n^2 + 3*n + 9, see A005471. - _R. J. Mathar_, Jun 06 2019
%F A027692 a(n) mod 6 = A109007(n+2). - _R. J. Mathar_, Jun 06 2019
%F A027692 Sum_{n>=0} 1/a(n) = Pi*tanh(Pi*3*sqrt(3)/2)/(3*sqrt(3)). - _Amiram Eldar_, Jan 17 2021
%F A027692 From _Elmo R. Oliveira_, Oct 30 2024: (Start)
%F A027692 E.g.f.: exp(x)*(7 + 2*x + x^2).
%F A027692 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%p A027692 A027692 := proc(n)
%p A027692     n*(n+1)+7 ;
%p A027692 end proc: # _R. J. Mathar_, Jun 06 2019
%t A027692 f[n_]:=n^2+n+7;f[Range[0,60]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 06 2011 *)
%o A027692 (PARI) a(n)=n^2+n+7 \\ _Charles R Greathouse IV_, Jun 11 2015
%o A027692 (GAP) List([0..50],n->n^2+n+7); # _Muniru A Asiru_, Jul 15 2018
%Y A027692 Cf. A002522, A005471 (subset of primes), A109007, A176271.
%K A027692 nonn,easy
%O A027692 0,1
%A A027692 _Patrick De Geest_