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A061035 Triangle T(m,n) = numerator of 1/m^2 - 1/n^2, n >= 1, m=n,n-1,n-2,...,1.

%I A061035 #17 Jun 04 2013 04:17:13
%S A061035 0,0,3,0,5,8,0,7,3,15,0,9,16,21,24,0,11,5,1,2,35,0,13,24,33,40,45,48,
%T A061035 0,15,7,39,3,55,15,63,0,17,32,5,56,65,8,77,80,0,19,9,51,4,3,21,91,6,
%U A061035 99,0,21,40,57,72,85,96,105,112,117,120,0,23,11,7,5,95,1,119,1,5,35,143,0
%N A061035 Triangle T(m,n) = numerator of 1/m^2 - 1/n^2, n >= 1, m=n,n-1,n-2,...,1.
%C A061035 Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).
%D A061035 J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 77.
%H A061035 Reinhard Zumkeller, <a href="/A061035/b061035.txt">Rows n=1..100 of triangle, flattened</a>
%H A061035 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 A061035 Eric Weisstein's World of Physics, <a href="http://scienceworld.wolfram.com/physics/BalmerFormula.html">Balmer Formula</a>
%e A061035 Triangle 1/m^2-1/n^2, m >= 1, 1<=n<=m, (i.e. with rows reversed) begins
%e A061035 0
%e A061035 3/4, 0
%e A061035 8/9, 5/36, 0
%e A061035 15/16, 3/16, 7/144, 0
%e A061035 24/25, 21/100, 16/225, 9/400, 0
%e A061035 35/36, 2/9, 1/12, 5/144, 11/900, 0
%t A061035 t[m_, n_] := Numerator[1/m^2 - 1/n^2]; Table[t[m, n], {n, 1, 12}, {m, n, 1, -1}] // Flatten(* _Jean-François Alcover_, Oct 17 2012 *)
%o A061035 (Haskell)
%o A061035 import Data.Ratio ((%), numerator)
%o A061035 a061035 n k = a061035_tabl !! (n-1) !! (k-1)
%o A061035 a061035_row = map numerator . balmer where
%o A061035    balmer n = map (subtract (1 % n ^ 2) . (1 %) . (^ 2)) [n, n-1 .. 1]
%o A061035 a061035_tabl = map a061035_row [1..]
%o A061035 -- _Reinhard Zumkeller_, Apr 12 2012
%Y A061035 Cf. A061036. Rows give A061037-A061050.
%Y A061035 Cf. A126252.
%K A061035 nonn,tabl,easy,nice,frac
%O A061035 1,3
%A A061035 _N. J. A. Sloane_, May 26 2001
%E A061035 More terms from _Naohiro Nomoto_, Jul 15 2001