%I #184 Feb 28 2024 05:31:19

%S 0,5,12,21,32,45,60,77,96,117,140,165,192,221,252,285,320,357,396,437,

%T 480,525,572,621,672,725,780,837,896,957,1020,1085,1152,1221,1292,

%U 1365,1440,1517,1596,1677,1760,1845,1932,2021,2112,2205,2300,2397,2496,2597

%N a(n) = n^2 - 4.

%C Nonnegative X values of solutions to the equation X^3 + 4*X^2 = Y^2. The respective Y values are n*(n^2 - 4). - _Mohamed Bouhamida_, Nov 06 2007

%C Discriminants of binary forms x^2 + n*x*y + y^2 (for n > 1). - _Artur Jasinski_, Apr 28 2008

%C a(n)*a(n-1) + 4 = (a(n)-n)^2. This is the case d = 4 in the general (n^2-d)*((n-1)^2-d) + d = (n^2-n-d)^2. - _Bruno Berselli_, Dec 07 2011

%C Interleaving of A134582 and A078371. - _Bruce J. Nicholson_, Oct 14 2019

%D Alain Connes, Noncommutative Geometry, Academic Press, 1994, p. 35.

%H G. C. Greubel, <a href="/A028347/b028347.txt">Table of n, a(n) for n = 2..1000</a>

%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.

%H F. P. Muga II, <a href="https://www.researchgate.net/publication/267327689_Extending_the_Golden_Ratio_and_the_Binet-de_Moivre_Formula">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, March 2014; Preprint on ResearchGate.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hydrogen_spectral_series">Hydrogen spectral series</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F Except for initial term, denominators of energies of hydrogen lines.

%F a(n+2) = n*(n+4). G.f.: x^3*(5-3*x)/(1-x)^3. - _Barry E. Williams_, Jun 16 2000, _R. J. Mathar_, Aug 06 2009

%F a(n) = 2*n + a(n-1) - 1. - _Vincenzo Librandi_, Aug 02 2010

%F Sum_{n >= 3} 1/a(n) = 25/48 = 0.52083333... = 100*A021196. - _R. J. Mathar_, Mar 22 2011

%F a(n) = x, the solution of k = (sqrt(x)+n)/2 and k + (1/k) = n (also valid for a(0) = -4 and a(1) = -3). - _Charles L. Hohn_, Apr 16 2011

%F E.g.f.: (x^2 + x - 4)*exp(x). - _G. C. Greubel_, Jul 17 2017

%F Sum_{n>=3} (-1)^(n+1)/a(n) = 7/48. - _Amiram Eldar_, Jul 03 2020

%F From _Amiram Eldar_, Feb 05 2024: (Start)

%F Product_{n>=3} (1 - 1/a(n)) = 6*sin(sqrt(5)*Pi)/(sqrt(5)*Pi).

%F Product_{n>=3} (1 + 1/a(n)) = -4*sqrt(3)*sin(sqrt(3)*Pi)/Pi. (End)

%e G.f. = 5*x^3 + 12*x^4 + 21*x^5 + 32*x^6 + 45*x^7 + 60*x^8 + 77*x^9 + 96*x^10 + ...

%p A028347:=n->n^2 - 4; seq(A028347(n), n=2..100); # _Wesley Ivan Hurt_, Mar 11 2014

%t Table[n^2 - 4, {n, 2, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 06 2008 *)

%t LinearRecurrence[{3,-3,1}, {0, 5, 12}, 50] (* _G. C. Greubel_, Nov 25 2016 *)

%o (PARI) a(n)=n^2-4 \\ _Charles R Greathouse IV_, Mar 11 2014

%Y a(n), n>=3, second column (used for the Balmer series of the hydrogen atom) of triangle A120070.

%Y Cf. A005563, A046092, A001082, A002378, A036666, A062717, A078371, A134582.

%K nonn,easy

%O 2,2

%A _N. J. A. Sloane_