A061039 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,2

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).

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

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 *)

PROG

(Haskell)

import Data.Ratio ((%), numerator)

a061039 n = numerator $ 1%9 - 1%n ^ 2 -- Reinhard Zumkeller, Jan 03 2012

(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