A028347 a(n) = n^2 - 4.

0, 5, 12, 21, 32, 45, 60, 77, 96, 117, 140, 165, 192, 221, 252, 285, 320, 357, 396, 437, 480, 525, 572, 621, 672, 725, 780, 837, 896, 957, 1020, 1085, 1152, 1221, 1292, 1365, 1440, 1517, 1596, 1677, 1760, 1845, 1932, 2021, 2112, 2205, 2300, 2397, 2496, 2597

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Discriminants of binary forms x^2 + n*x*y + y^2 (for n > 1). - Artur Jasinski, Apr 28 2008

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

Interleaving of A134582 and A078371. - Bruce J. Nicholson, Oct 14 2019

Examples

			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 + ...
		

References

• Alain Connes, Noncommutative Geometry, Academic Press, 1994, p. 35.

Links

• G. C. Greubel, Table of n, a(n) for n = 2..1000
• Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
• F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
• Eric Weisstein's World of Mathematics, Near-Square Prime.
• Wikipedia, Hydrogen spectral series.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

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

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

Programs

Maple

A028347:=n->n^2 - 4; seq(A028347(n), n=2..100); # Wesley Ivan Hurt, Mar 11 2014

Mathematica

Table[n^2 - 4, {n, 2, 100}] (* Vladimir Joseph Stephan Orlovsky, Nov 06 2008 *)
LinearRecurrence[{3,-3,1}, {0, 5, 12}, 50] (* G. C. Greubel, Nov 25 2016 *)

PARI

a(n)=n^2-4 \\ Charles R Greathouse IV, Mar 11 2014

Formula

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

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

a(n) = 2*n + a(n-1) - 1. - Vincenzo Librandi, Aug 02 2010

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

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

E.g.f.: (x^2 + x - 4)*exp(x). - G. C. Greubel, Jul 17 2017

Sum_{n>=3} (-1)^(n+1)/a(n) = 7/48. - Amiram Eldar, Jul 03 2020

From Amiram Eldar, Feb 05 2024: (Start)

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

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