A061039 Numerator of 1/9 - 1/n^2.

0, 7, 16, 1, 40, 55, 8, 91, 112, 5, 160, 187, 8, 247, 280, 35, 352, 391, 16, 475, 520, 7, 616, 667, 80, 775, 832, 11, 952, 1015, 40, 1147, 1216, 143, 1360, 1435, 56, 1591, 1672, 65, 1840, 1927, 224, 2107, 2200, 85, 2392, 2491, 32, 2695, 2800, 323, 3016, 3127

Offset: 3

Author

N. J. A. Sloane, May 26 2001

Keywords

• nonn
• frac
• nice
• easy

Comments

The denominators are given in A061040.

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

References

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 3..1000
• J. J. O'Connor and E. F. Robertson, Johannes Robert Rydberg
• Eric Weisstein's World of Physics, Balmer Formula
• Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Crossrefs

Cf. A061035..A061050.

Programs

Haskell

import Data.Ratio ((%), numerator)
a061039 n = numerator $ 1%9 - 1%n ^ 2  -- Reinhard Zumkeller, Jan 03 2012

Maple

A061039:=n->numer(1/9-1/n^2): seq(A061039(n), n=3..80); # Wesley Ivan Hurt, Apr 12 2017

Mathematica

Table[Numerator[1/9 - 1/n^2], {n, 3, 60}] (* Stefan Steinerberger, Apr 16 2006 *)

PARI

a(n)=numerator(1/9-1/n^2) \\ Charles R Greathouse IV, Nov 20 2012

Python

from math import gcd
def A061039(n): return (n**2-9)//gcd(n**2-9,9*n**2) # Chai Wah Wu, Apr 02 2021

Formula

a(n) <= n^2 - 9; if n is not divisible by 3 then a(n) = n^2 - 9. - Stefan Steinerberger, Apr 16 2006

a(n) = 3*a(n-27) - 3*a(n-54) + a(n-81) for n > 83. - Colin Barker, Oct 09 2016

a(n) = (n^2 - 9)/9^2 if n == 3 or 24 (mod 27), a(n) = (n^2 - 9)/(3*9) if n == 6 or 24 or 15 or 21 (mod 27), a(n) = (n^2 - 9)/9 if n == 0 (mod 9) and n^2 - 9 otherwise. From the period length 27 sequence gcd(n^2 - 9, 9*n^2). - Wolfdieter Lang, Mar 15 2018