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A120070 Triangle of numbers used to compute the frequencies of the spectral lines of the hydrogen atom. 43
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

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.

Frenicle wrote this as a(n+1) = A140978(n) - A133819(n-1). - Paul Curtz, Aug 19 2008

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

REFERENCES

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.

LINKS

Stanislav Sykora, Table of n, a(n) for n = 2..79801

W. Lang: First ten rows and more.

M. de Frenicle, Methode pour trouver la solutions des problemes par les exclusions, in: Divers ouvrages des mathematiques et de physique par messieurs de l'academie royale des sciences, (1693) pp 1-44, page 11.

Wikipedia, Spin (physics)

Wikipedia, Hydrogen spectral series

FORMULA

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

EXAMPLE

Triangle begins

  [ 3];

  [ 8, 5];

  [15, 12,  7];

  [24, 21, 16,  9];

  [35, 32, 27, 20, 11];

  ...

MATHEMATICA

ColumnForm[Table[n^2 - k^2, {n, 2, 13}, {k, n - 1}], Center] (* Alonso del Arte, Oct 26 2011 *)

PROG

(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), ", ")));

\\ Joerg Arndt, Feb 23 2014

CROSSREFS

Row sums give A016061(n-1), n>=2.

Cf. A120072/A120073 numerator and denominator tables for rationals r(m, n).

Sequence in context: A050093 A120072 A166492 * A143753 A121164 A086872

Adjacent sequences:  A120067 A120068 A120069 * A120071 A120072 A120073

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Jul 20 2006

STATUS

approved

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Last modified January 22 06:25 EST 2019. Contains 319353 sequences. (Running on oeis4.)