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A028347 a(n) = n^2 - 4. 56
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Sequence allows us to find X values of the equation: X^3 + 4*X^2 = Y^2. To find Y values: b(n)=n*(n^2 - 4). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007

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

Number of units of a(n) belongs to a periodic sequence: 0, 5, 2, 1, 2, 5, 0, 7, 6, 7. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009

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

A. Connes, Noncommutative Geometry, Academic Press, 1994, p. 35.

F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.

LINKS

G. C. Greubel, Table of n, a(n) for n = 2..1000

M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550, 2013

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

a(n) = 5*binomial(n+1, 2) - 3*binomial(n, 2) for n = 0, 1, ... - Felix P. Muga II, Mar 11 2014

a(n) = a(n-1) + 2*n + 3 with a(0)=0 and a(1)=5. - Eric Desbiaux, Nov 21 2016

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

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 *)

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.

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

Sequence in context: A272012 A272451 A097984 * A256320 A038794 A225284

Adjacent sequences:  A028344 A028345 A028346 * A028348 A028349 A028350

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 30 03:42 EDT 2017. Contains 287305 sequences.