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Search: id:A120070
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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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45
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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
(list; table; graph; listen)
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OFFSET
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2,1
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COMMENT
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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.
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
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REFERENCES
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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.
(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
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LINKS
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W. Lang: First ten rows and more.
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FORMULA
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a(m,n)= m^2 - n^2 for m-1>=n, else 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.
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EXAMPLE
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[3];[8,5];[15,12,7];[24,21,16,9];...
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CROSSREFS
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Row sums give A016061(n-1), n>=2.
Cf. A120072/A120073 numerator and denominator tables for rationals r(m, n).
Sequence in context: A120072 A166492 A143813 this_sequence A143753 A121164 A086872
Adjacent sequences: A120067 A120068 A120069 this_sequence A120071 A120072 A120073
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 20 2006
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