cp's OEIS Frontend

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

%I A061039 #47 Jul 29 2023 07:13:48
%S A061039 0,7,16,1,40,55,8,91,112,5,160,187,8,247,280,35,352,391,16,475,520,7,
%T A061039 616,667,80,775,832,11,952,1015,40,1147,1216,143,1360,1435,56,1591,
%U A061039 1672,65,1840,1927,224,2107,2200,85,2392,2491,32,2695,2800,323,3016,3127
%N A061039 Numerator of 1/9 - 1/n^2.
%C A061039 The denominators are given in A061040.
%C A061039 From Paschen spectrum of hydrogen. Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).
%D A061039 J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 78.
%H A061039 Reinhard Zumkeller, <a href="/A061039/b061039.txt">Table of n, a(n) for n = 3..1000</a>
%H A061039 J. J. O'Connor and E. F. Robertson, <a href="http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Rydberg.html">Johannes Robert Rydberg</a>
%H A061039 Eric Weisstein's World of Physics, <a href="http://scienceworld.wolfram.com/physics/BalmerFormula.html">Balmer Formula</a>
%H A061039 <a href="/index/Rec#order_81">Index entries for linear recurrences with constant coefficients</a>, 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).
%F A061039 a(n) <= n^2 - 9; if n is not divisible by 3 then a(n) = n^2 - 9. - _Stefan Steinerberger_, Apr 16 2006
%F A061039 a(n) = 3*a(n-27) - 3*a(n-54) + a(n-81) for n > 83. - _Colin Barker_, Oct 09 2016
%F A061039 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
%p A061039 A061039:=n->numer(1/9-1/n^2): seq(A061039(n), n=3..80); # _Wesley Ivan Hurt_, Apr 12 2017
%t A061039 Table[Numerator[1/9 - 1/n^2], {n, 3, 60}] (* _Stefan Steinerberger_, Apr 16 2006 *)
%o A061039 (Haskell)
%o A061039 import Data.Ratio ((%), numerator)
%o A061039 a061039 n = numerator $ 1%9 - 1%n ^ 2  -- _Reinhard Zumkeller_, Jan 03 2012
%o A061039 (PARI) a(n)=numerator(1/9-1/n^2) \\ _Charles R Greathouse IV_, Nov 20 2012
%o A061039 (Python)
%o A061039 from math import gcd
%o A061039 def A061039(n): return (n**2-9)//gcd(n**2-9,9*n**2) # _Chai Wah Wu_, Apr 02 2021
%Y A061039 Cf. A061035..A061050.
%K A061039 nonn,frac,nice,easy
%O A061039 3,2
%A A061039 _N. J. A. Sloane_, May 26 2001