%I A120070
%S A120070 3,8,5,15,12,7,24,21,16,9,35,32,27,20,11,48,45,40,33,24,13,63,60,55,48,
%T A120070 39,28,15,80,77,72,65,56,45,32,17,99,96,91,84,75,64,51,36,19,120,117,
%U A120070 112,105,96,85,72,57,40,21
%N A120070 Triangle of numbers used to compute the frequencies of the spectral lines 
               of the hydrogen atom.
%C A120070 The rationals r(m,n):=a(m,n)/(m^2*n^2), for m-1>=n, else 0, are used 
               to compute the frequencies of the spectral lines of the H-atom according 
               to quantum theory: nu(m,n) = r(m,n)*c*R' with c*R'=3.287*10^15 s^(-1) 
               an approximation for the Rydberg frequency. R' indicates, that the 
               correction factor 1/(1+m_e/m_p), approximately 0.9995, with the masses 
               for the electron and proton, has been used for the Rydberg constant 
               R_infinity. c:=299792458 m/s is, per definition, the velocity of 
               light in vacuo (see A003678).
%C A120070 In order to compute the wave length of the spectral lines approximately 
               one uses the reciprocal rationals: lambda(m,n):= c/nu(m,n) = (1/r(m,
               n))*91.1961 nm. 1 nm = 10^{-9} m. For the corresponding energies 
               one uses approximately E(m,n)= r(m,n)*13.599 eV (electron Volts).
%C A120070 The author was inspired by Dewdney's book to compile this table and related 
               ones.
%C A120070 For the approximate frequencies, energies and wavelengths of the first 
               members of the Lyman (n=1,m>=2), Balmer (n=2,m>=3), Paschen (n=3,
               m>=4), Brackett (n=4,m>=5) and Pfund (n=5,m>=6) series see the W. 
               Lang link under A120072.
%C A120070 Based on Frenicle's b(n)= 4, 9, 9, 16, 16, 16, 25, 25, 25, 25, ... and 
               c(n)= 1, 1, 4, 1, 4, 9, 1, 4, 9, 16, 1, 4, 9, 16, 25, ... =A133819: 
               a(n)=b(n)-c(n). - Paul Curtz (bpcrtz(AT)free.fr), Aug 19 2008
%D A120070 A. K. Dewdney, Reise in das Innere der Mathematik, Birkhaeuser, Basel, 
               2000, pp. 148-154; engl.: A Mathematical Mystery Tour, John Wiley 
               & Sons, N.Y., 1999.
%D A120070 (Bernard) de? Frenicle (de Bessy), studying Pythagorean triangles: Methode 
               pour trouver ...; in Divers ouvrages de mathematique et de physique 
               par Messieurs de l'Academie Royale des Sciences, In-folio, (4)+6+519 
               pages, Paris, 1693. - Paul Curtz (bpcrtz(AT)free.fr), Aug 18 2008
%H A120070 W. Lang: <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A120070.text">
               First ten rows and more.</a>
%F A120070 a(m,n)= m^2 - n^2 for m-1>=n, else 0.
%F A120070 G.f. for column n=1,2,...: x^(n+1)*((2*n+1)- (2*n-1)*x)/(1-x)^3.
%F A120070 G.f. for rationals r(m,n), n=1,2,...,10 see W. Lang link.
%e A120070 [3];[8,5];[15,12,7];[24,21,16,9];...
%Y A120070 Row sums give A016061(n-1), n>=2.
%Y A120070 Cf. A120072/A120073 numerator and denominator tables for rationals r(m, 
               n).
%Y A120070 Sequence in context: A120072 A166492 A143813 this_sequence A143753 A121164 
               A086872
%Y A120070 Adjacent sequences: A120067 A120068 A120069 this_sequence A120071 A120072 
               A120073
%K A120070 nonn,easy,tabl
%O A120070 2,1
%A A120070 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 20 
               2006