OFFSET
2,2
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
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).
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)
EXAMPLE
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 + ...
MAPLE
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 *)
PROG
(PARI) a(n)=n^2-4 \\ Charles R Greathouse IV, Mar 11 2014
CROSSREFS
a(n), n>=3, second column (used for the Balmer series of the hydrogen atom) of triangle A120070.
KEYWORD
nonn,easy
AUTHOR
STATUS
approved