|
|
A120072
|
|
Numerator triangle for hydrogen spectrum rationals.
|
|
19
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,1
|
|
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.
|
|
LINKS
|
|
|
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.
|
|
EXAMPLE
|
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;
|
|
MATHEMATICA
|
|
|
PROG
|
(Magma) [Numerator(1/k^2 - 1/n^2): k in [1..n-1], n in [2..18]]; // G. C. Greubel, Apr 24 2023
(SageMath)
def A120072(n, k): return numerator(1/k^2 - 1/n^2)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|