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%I #50 Aug 28 2019 15:33:48
%S 3,8,5,15,12,7,24,21,16,9,35,32,27,20,11,48,45,40,33,24,13,63,60,55,
%T 48,39,28,15,80,77,72,65,56,45,32,17,99,96,91,84,75,64,51,36,19,120,
%U 117,112,105,96,85,72,57,40,21
%N Triangle of numbers used to compute the frequencies of the spectral lines of the hydrogen atom.
%C 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 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 The author was inspired by Dewdney's book to compile this table and related ones.
%C 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 Frenicle wrote this as a(n+1) = A140978(n) - A133819(n-1). - _Paul Curtz_, Aug 19 2008
%C This triangle also has an interpretation related to particle spin. For proper offset such that T(0,0) = 3, then, where h-bar = h/(2*Pi) = A003676/A019692 (= The Dirac constant, also known as Planck's reduced constant) and Spin(n/2) = h-bar/2*sqrt(n(n+2)), it follows that: h-bar/2*sqrt(T(r,k)) = h-bar/2*sqrt(T(r,0) - T(k-1,0)) = sqrt((Spin((r+1)/2))^2 - (Spin(k/2))^2). For example, for r = k = 4, then h-bar/2*sqrt(11) = h-bar/2*sqrt(T(4,4)) = h-bar/2*sqrt(T(4,0) - T(3,0)) = sqrt(h-bar^2/4*T(4,0) - h-bar^2/4*T(3,0)) = sqrt(h-bar^2/4*35 - h-bar^2/4*24) = sqrt((Spin((4+1)/2))^2 - (Spin(4/2))^2); 35 = 5*(5+2) & 24 = 4*(4+2). - _Raphie Frank_, Dec 30 2012
%D A. K. Dewdney, Reise in das Innere der Mathematik, Birkhäuser, Basel, 2000, pp. 148-154; engl.: A Mathematical Mystery Tour, John Wiley & Sons, N.Y., 1999.
%H Stanislav Sykora, <a href="/A120070/b120070.txt">Table of n, a(n) for n = 2..79801</a>
%H W. Lang: <a href="/A120070/a120070.txt">First ten rows and more.</a>
%H M. de Frenicle, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5493994j/">Methode pour trouver la solutions des problemes par les exclusions</a>, in: Divers ouvrages des mathematiques et de physique par messieurs de l'academie royale des sciences, (1693) pp 1-44, page 11.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Spin_(physics)">Spin (physics)</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hydrogen_spectral_series">Hydrogen spectral series</a>
%F a(m,n) = m^2 - n^2 for m-1 >= n, otherwise 0.
%F G.f. for column n=1,2,...: x^(n+1)*((2*n+1)- (2*n-1)*x)/(1-x)^3.
%F G.f. for rationals r(m,n), n=1,2,...,10 see W. Lang link.
%F T(r,k) = T(r,0) - T(k-1,0), T(0,0) = 3. - _Raphie Frank_, Dec 27 2012
%e Triangle begins
%e [ 3];
%e [ 8, 5];
%e [15, 12, 7];
%e [24, 21, 16, 9];
%e [35, 32, 27, 20, 11];
%e ...
%t ColumnForm[Table[n^2 - k^2, {n, 2, 13}, {k, n - 1}], Center] (* _Alonso del Arte_, Oct 26 2011 *)
%o (PARI) nmax=400;a=vector(1+nmax*(nmax-1)\2);idx=1;for(n=2,nmax,for(k=1,n-1,a[idx]=n*n-k*k;idx++)) \\ _Stanislav Sykora_, Feb 17 2014
%o (PARI) T(n,k)=n^2-k^2;
%o for (n=1,10,for(k=1,n-1, print1(T(n,k),", ")));
%o \\ _Joerg Arndt_, Feb 23 2014
%Y Row sums give A016061(n-1), n>=2.
%Y Cf. A120072/A120073 numerator and denominator tables for rationals r(m, n).
%K nonn,easy,tabl
%O 2,1
%A _Wolfdieter Lang_, Jul 20 2006
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