A028347 - OEIS
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
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.
FORMULA
Except for initial term, denominators of energies of hydrogen lines.
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
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 + ...
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 5, 12}, 50] (* G. C. Greubel, Nov 25 2016 *)
CROSSREFS
a(n), n>=3, second column (used for the Balmer series of the hydrogen atom) of triangle A120070.