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

3,2

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

Crossrefs

Cf. A061035..A061050.

Sequence in context: A087485 A156377 A069526 * A258771 A063593 A070417

Adjacent sequences: A061036 A061037 A061038 * A061040 A061041 A061042

Keyword

nonn,frac,nice,easy

Author

N. J. A. Sloane, May 26 2001

Status

approved