A120072 Numerator triangle for hydrogen spectrum rationals.

3, 8, 5, 15, 3, 7, 24, 21, 16, 9, 35, 2, 1, 5, 11, 48, 45, 40, 33, 24, 13, 63, 15, 55, 3, 39, 7, 15, 80, 77, 8, 65, 56, 5, 32, 17, 99, 6, 91, 21, 3, 4, 51, 9, 19, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21, 143, 35, 5, 1, 119, 1, 95, 5, 7, 11, 23

Offset: 2

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• tabl
• frac

Comments

Frequencies or energies of the spectral lines of the hydrogen (H) atom are given, according to quantum theory, by r(m,n)*3.287*PHz (1 Peta Hertz= 10^15 s^{-1}) or r(m,n)*13.599 eV (electron Volts), respectively. The wave lengths are lambda(m,n) = (1/r(m,n))* 91.196 nm (all decimals rounded). See the W. Lang link for more details.

The spectral series for n=1,2,...,7, m>=n+1, are named after Lyman, Balmer, Paschen, Brackett, Pfund, Humphreys, Hansen-Strong, respectively.

The corresponding denominator triangle is A120073.

The rationals are r(m,n):= a(m,n)/A120073(m,n) = A120070(m,n)/(m^2*n^2) = 1/ n^2 - 1/m^2 and they are given in lowest terms.

Examples

			For the rational triangle see W. Lang link.
Numerator triangle begins as:
   3;
   8,  5;
  15,  3,  7;
  24, 21, 16,  9;
  35,  2,  1,  5, 11;
  48, 45, 40, 33, 24, 13;
  63, 15, 55,  3, 39,  7, 15;
  80, 77,  8, 65, 56,  5, 32, 17;
  99,  6, 91, 21,  3,  4, 51,  9, 19;
		

Links

• G. C. Greubel, Rows n = 2..50 of the triangle, flattened
• Wolfdieter Lang, First ten rows, rationals and more.
• T. Lyman, The Spectrum of Hydrogen in the Region of Extremely Short Wave-Lengths, The Astrophysical Journal, 23 (April 1906), 181-210. - _Paul Curtz_, May 30 2017

Crossrefs

Row sums give A120074.

Row sums of r(m, n) triangle give A120076(m)/A120077(m), m>=2.

Cf. A120070, A120073, A120075, A126252.

Programs

Magma

[Numerator(1/k^2 - 1/n^2): k in [1..n-1], n in [2..18]]; // G. C. Greubel, Apr 24 2023

Mathematica

Table[1/n^2 - 1/m^2, {m,2,12}, {n,m-1}]//Flatten//Numerator (* Jean-François Alcover, Sep 16 2013 *)

SageMath

def A120072(n,k): return numerator(1/k^2 - 1/n^2)
flatten([[A120072(n,k) for k in range(1,n)] for n in range(2,19)]) # G. C. Greubel, Apr 24 2023

Formula

a(m,n) = numerator(r(m,n)) with r(m,n) = 1/n^2 - 1/m^2, m>=2, n=1..m-1.

The g.f.s for the columns n=1,..,10 of triangle r(m,n) = a(m, n) / A120073(m, n), m >= 2, 1 <= n <= m-1, are given in the W. Lang link.