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A061041 Numerator of 1/16 - 1/n^2.

%I A061041 #52 Apr 20 2023 02:21:57
%S A061041 0,9,5,33,3,65,21,105,1,153,45,209,15,273,77,345,3,425,117,513,35,609,
%T A061041 165,713,3,825,221,945,63,1073,285,1209,5,1353,357,1505,99,1665,437,
%U A061041 1833,15,2009,525,2193,143,2385,621,2585,21,2793,725
%N A061041 Numerator of 1/16 - 1/n^2.
%C A061041 From Brackett spectrum of hydrogen. Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).
%D A061041 J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 78.
%H A061041 Reinhard Zumkeller, <a href="/A061041/b061041.txt">Table of n, a(n) for n = 4..10000</a>
%H A061041 Frederick Sumner Brackett, <a href="http://adsabs.harvard.edu/abs/1922ApJ....56..154B">Visible and Infra-Red Radiation of Hydrogen</a> Astrophysical Journal, 56, No. 3 (1922) pp. 154-161.
%H A061041 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 A061041 Eric Weisstein's World of Physics, <a href="http://scienceworld.wolfram.com/physics/BalmerFormula.html">Balmer Formula</a>
%H A061041 <a href="/index/Rec#order_72">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,3,0,0,0,0,0,0,0, -6,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0, -10,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,-3,0,0,0,0,0,0,0,1).
%F A061041 a(4*n+6) = A078371(n). - _Paul Curtz_, Oct 05 2008
%F A061041 a(n) = 3*a(n-8) - 6*a(n-16) + 10*a(n-24) - 12*a(n-32) + 12*a(n-40) - 10*a(n-48) + 6*a(n-56) - 3*a(n-64) + a(n-72). - _Charles R Greathouse IV_, Aug 17 2011
%F A061041 a(n) = (n^2-16) / gcd(16*n^2, n^2-16). - _Franklin T. Adams-Watters_, Sep 25 2011, corrected by _Colin Barker_, Jan 13 2014.
%p A061041 A061041:=n->numer(1/16 - 1/n^2); seq(A061041(n), n=4..60); # _Wesley Ivan Hurt_, Jan 28 2014
%t A061041 Numerator/@(1/16-1/Range[4,60]^2)  (* _Harvey P. Dale_, Mar 24 2011 *)
%o A061041 (PARI) a(n)=numerator(1/16-1/n^2) \\ _Charles R Greathouse IV_, Aug 17 2011
%o A061041 (Haskell)
%o A061041 import Data.Ratio ((%), numerator)
%o A061041 a061041 n = numerator (1%16 - 1%n^2)  -- _Reinhard Zumkeller_, May 30 2012
%o A061041 (Magma) [Numerator(1/16 -1/n^2): n in [4..60]]; // _G. C. Greubel_, Apr 19 2023
%o A061041 (SageMath) [numerator(1/16 -1/n^2) for n in range(4,61)] # _G. C. Greubel_, Apr 19 2023
%Y A061041 Cf. A061035 - A061050, A064038, A078371.
%K A061041 nonn,frac,nice,easy
%O A061041 4,2
%A A061041 _N. J. A. Sloane_, May 26 2001