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A061036 Triangle T(m,n) = denominator of 1/m^2 - 1/n^2, n >= 1, m=n,n-1,n-2,...,1.

%I A061036 #16 Jul 13 2013 12:02:38
%S A061036 1,1,4,1,36,9,1,144,16,16,1,400,225,100,25,1,900,144,12,9,36,1,1764,
%T A061036 1225,784,441,196,49,1,3136,576,1600,64,576,64,64,1,5184,3969,324,
%U A061036 2025,1296,81,324,81,1,8100,1600,4900,225,100,400,900,25,100,1,12100,9801
%N A061036 Triangle T(m,n) = denominator of 1/m^2 - 1/n^2, n >= 1, m=n,n-1,n-2,...,1.
%C A061036 Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).
%D A061036 J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 77.
%H A061036 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 A061036 Eric Weisstein's World of Physics, <a href="http://scienceworld.wolfram.com/physics/BalmerFormula.html">Balmer Formula</a>
%H A061036 Reinhard Zumkeller, <a href="/A061036/b061036.txt">Rows n=..100 of triangle, flattened</a>
%e A061036 Triangle 1/m^2-1/n^2, m >= 1, 1<=n<=m, (i.e. with rows reversed) begins
%e A061036 0
%e A061036 3/4, 0
%e A061036 8/9, 5/36, 0
%e A061036 15/16, 3/16, 7/144, 0
%e A061036 24/25, 21/100, 16/225, 9/400, 0
%e A061036 35/36, 2/9, 1/12, 5/144, 11/900, 0
%t A061036 t[m_, n_] := Denominator[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 A061036 (Haskell)
%o A061036 import Data.Ratio ((%), denominator)
%o A061036 a061036 n k = a061036_tabl !! (n-1) !! (k-1)
%o A061036 a061036_row = map denominator . balmer where
%o A061036    balmer n = map (subtract (1 % n ^ 2) . (1 %) . (^ 2)) [n, n-1 .. 1]
%o A061036 a061036_tabl = map a061036_row [1..]
%o A061036 -- _Reinhard Zumkeller_, Apr 12 2012
%Y A061036 Cf. A061035. Rows give A061037-A061050.
%Y A061036 Cf. A126252.
%K A061036 nonn,tabl,easy,nice,frac
%O A061036 1,3
%A A061036 _N. J. A. Sloane_, May 26 2001
%E A061036 More terms from _Naohiro Nomoto_, Jul 15 2001