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Wavelength (in ångströms) of the series limit of the Hydrogen spectrum for main quantum number n.
2

%I #24 May 29 2021 05:47:58

%S 911,3645,8201,14580,22782,32806,44652,58321,73813,91127,110263,

%T 131222,154004

%N Wavelength (in ångströms) of the series limit of the Hydrogen spectrum for main quantum number n.

%C The series for n=1, 2, etc. are named after Lyman, Balmer, etc.

%C The sequence is ill-defined since it depends on the incompletely known natural constant A081821. Although the sequence is essentially proportional to n^2, rounding effects prevent high-term predictability of the differences. Taking relativistic corrections into account will worsen the situation. - _R. J. Mathar_, Oct 17 2008

%D Robert Duane Cowan, The Theory of Atomic Structure and Spectra, Univ. California Press, 1981; in 3-7 Relativistic corrections.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Rydberg_formula">Rydberg Formula</a>.

%F a(n+1) - a(n) = A145647(n), first differences.

%F a(n) = round(n^2*10^10/A081821). - _R. J. Mathar_, Oct 17 2008

%F Conjectures from _Colin Barker_, May 07 2019: (Start)

%F G.f.: x*(911 + 912*x + 910*x^2 + 913*x^3 - x^4) / ((1 - x)^3*(1 + x^2)).

%F a(n) = (-1/8+i/8)*((3+3*i) + (-i)^n + i^(1+n) - (1+i)*n - (3645+3645*i)*n^2) where i=sqrt(-1).

%F a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>5.

%F (End)

%p R := 10973731.568527 : for n from 1 to 13 do printf("%d,",round(n^2*1.0e10/R) ); od; # _R. J. Mathar_, Oct 17 2008

%K nonn,less

%O 1,1

%A _Paul Curtz_, Oct 15 2008

%E Edited by _R. J. Mathar_, Oct 17 2008