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A120070
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Triangle of numbers used to compute the frequencies of the spectral lines of the hydrogen atom.
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49
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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, 39, 28, 15, 80, 77, 72, 65, 56, 45, 32, 17, 99, 96, 91, 84, 75, 64, 51, 36, 19, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21
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OFFSET
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2,1
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COMMENTS
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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).
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).
The author was inspired by Dewdney's book to compile this table and related ones.
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.
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
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REFERENCES
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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.
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LINKS
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FORMULA
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a(m,n) = m^2 - n^2 for m-1 >= n, otherwise 0.
G.f. for column n=1,2,...: x^(n+1)*((2*n+1)- (2*n-1)*x)/(1-x)^3.
G.f. for rationals r(m,n), n=1,2,...,10 see W. Lang link.
T(r,k) = T(r,0) - T(k-1,0), T(0,0) = 3. - Raphie Frank, Dec 27 2012
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EXAMPLE
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Triangle begins
[ 3];
[ 8, 5];
[15, 12, 7];
[24, 21, 16, 9];
[35, 32, 27, 20, 11];
...
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MATHEMATICA
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ColumnForm[Table[n^2 - k^2, {n, 2, 13}, {k, n - 1}], Center] (* Alonso del Arte, Oct 26 2011 *)
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PROG
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(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
(PARI) T(n, k)=n^2-k^2;
for (n=1, 10, for(k=1, n-1, print1(T(n, k), ", ")));
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CROSSREFS
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Cf. A120072/A120073 numerator and denominator tables for rationals r(m, n).
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KEYWORD
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AUTHOR
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STATUS
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approved
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