A061037 Numerator of 1/4 - 1/n^2.

0, 5, 3, 21, 2, 45, 15, 77, 6, 117, 35, 165, 12, 221, 63, 285, 20, 357, 99, 437, 30, 525, 143, 621, 42, 725, 195, 837, 56, 957, 255, 1085, 72, 1221, 323, 1365, 90, 1517, 399, 1677, 110, 1845, 483, 2021, 132, 2205, 575, 2397, 156, 2597, 675

Offset: 2

Author

N. J. A. Sloane, May 26 2001

Keywords

• nonn
• frac
• nice
• easy

Comments

From Balmer spectrum of hydrogen. Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).

a(-2) = 0, a(-1) = a(1) = -3. - Paul Curtz, Feb 19 2011

Can be thought of as 4 interlocking sequences, each of the form a(n) = 3a(n - 1) - 3a(n - 2) + a(n - 3). - Charles R Greathouse IV, May 27 2011

References

• J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 78.

Links

• Harry J. Smith, Table of n, a(n) for n=2..1000
• J. J. O'Connor and E. F. Robertson, Johannes Robert Rydberg.
• Wikipedia, Balmer series.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Crossrefs

Cf. A061038 (denominators), A061035-A061050, A126252, A028347.

Programs

Haskell

import Data.Ratio ((%), numerator)
a061037 n = numerator (1%4 - 1%n^2)  -- Reinhard Zumkeller, Dec 17 2011

Magma

[ Numerator(1/4-1/n^2): n in [2..52] ]; // Bruno Berselli, Feb 10 2011

Mathematica

f[n_] := n/GCD[n, 4]; Array[f[#] f[# + 4] &, 51, 0]
f[n_] := Numerator[(n - 2) (n + 2)/(4 n^2)]; Array[f, 51, 2] (* Or *)
a[n_] := 3 a[n - 4] - 3 a[n - 8] + a[n - 12]; a[1] = -3; a[2] = 0; a[3] = 5; a[4] = 3; a[5] = 21; a[6] = 2; a[7] = 45; a[8] = 15; a[9] = 77; a[10] = 6; a[11] = 117; a[12] = 35; Array[a, 51, 2] (* Robert G. Wilson v *)
Numerator[1/4-1/Range[2,60]^2] (* Harvey P. Dale, Aug 18 2011 *)

PARI

a(n) = { numerator(1/4 - 1/n^2) } \\ Harry J. Smith, Jul 17 2009

Formula

G.f.: x^2(-3x^11-x^10-3x^9+14x^7+6x^6+30x^5+2x^4+21x^3+3x^2+5x)/(1-x^4)^3.

a(4n+2) = n(n+1), a(2n+3) = (2n+1)(2n+5), a(4n+4) = (2n+1)(2n+3). - Ralf Stephan, Jun 10 2005

a(n+2) = A060819(n) * A060819(n+4).

a(n) = (n^2-4)*(3*i^n+3*(-i)^n-27*(-1)^n+37)/64, where i is the imaginary unit. - Bruno Berselli, Feb 10 2011

a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12). - Paul Curtz, Feb 28 2011

a(n+2) = n*(n+4)/(period 4: 16, 1, 4, 1 = A146160(n)) = A028347(n+2) / A146160(n). - Paul Curtz, Mar 24 2011 [edited by Franklin T. Adams-Watters, Mar 25 2011]

a(n) = (n^2-4) / gcd(4*n^2, (n^2-4)). - Colin Barker, Jan 13 2014

Sum_{n>=3} 1/a(n) = 11/6. - Amiram Eldar, Aug 12 2022