%I #8 Dec 06 2014 18:11:59

%S 1523310,2056410,2303240,2437290,2518130

%N Wavenumbers of red, turquoise, blue, indigo and violet in the spectrum of hydrogen, as first measured by Robert Bunsen and Gustav Kirchhoff in 1859.

%C How Johann Jakob Ballmer found his formula in 1885 by analyzing and manipulating the ratios of these data:

%C r(1) = a(1)/a(1) = 1,

%C a(2)/a(1) = 1.349961..., rounded: r(2) = 135/100 = 27/20,

%C a(3)/a(1) = 1.511996..., rounded: r(3) = 1512/1000 = 189/125,

%C a(4)/a(1) = 1.599996..., rounded: r(4) = 16/10 = 8/5,

%C a(5)/a(1) = 1.6530647..., r(5) = 81/49 = 2-1/(3-1/(9-1/2)), derived from a(5)/a(1) = 2-1/(3-1/(9-3095/6216)) when replacing 3095/6216 by 1/2;

%C the multiplication of these fractions by 5/36 is the key trick to get more handy figures to see eventually increasing squares in the denominators by an appropriate expansion:

%C b(1) = r(1)*5/36 = 5 / 36,

%C b(2) = r(2)*5/36 = 3 / 16,

%C b(3) = r(3)*5/36 = 21 / 100,

%C b(4) = r(4)*5/36 = 2 / 9,

%C b(5) = r(5)*5/36 = 45 / 196;

%C ... b(1) .|.... b(2) ..|.... b(3) ..|.... b(4) ..|.... b(5),

%C ... 5/36 .|.... 3/16 ..|... 21/100 .|.... 2/9 ...|... 45/196,

%C ... 5/36 .|... 12/64 ..|... 21/100 .|... 32/144 .|... 45/196,

%C (9-4)/9*4 |(16-4)/16*4 |(25-4)/25*4 |(36-4)/36*4 |(49-4)/49*4,

%C this last step was the crowning achievement: the discovery of the pattern (x-y)/x*y,

%C b(n) = ((n+2)^2 - 4)/(4*(n+2)^2) = 1/4 - 1/(n+2)^2;

%C 1<=n<=5: b(n) = A061037(n+2)/A061038(n+2) = A120072(n+2,2)/A120073(n+2,2).

%D R. Taschner, Der Zahlen gigantischer Schatten, Vieweg 2005, 137-143.

%H Science Trek, <a href="http://www.colorado.edu/physics/2000/quantumzone/balmer.html">Balmer Formula</a>

%H Eric Weisstein's World of Physics, <a href="http://scienceworld.wolfram.com/physics/BalmerFormula.html">Balmer Formula</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Empirical_formula">Empirical formula</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Johann_Jakob_Balmer">Johann Jakob Balmer</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bunsen">Robert Wilhelm Bunsen</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Gustav_Kirchhoff">Gustav Robert Kirchhoff</a>

%K nonn,fini,full

%O 1,1

%A _Reinhard Zumkeller_, Dec 22 2006