A061036 Triangle T(m,n) = denominator of 1/m^2 - 1/n^2, n >= 1, m=n,n-1,n-2,...,1.

1, 1, 4, 1, 36, 9, 1, 144, 16, 16, 1, 400, 225, 100, 25, 1, 900, 144, 12, 9, 36, 1, 1764, 1225, 784, 441, 196, 49, 1, 3136, 576, 1600, 64, 576, 64, 64, 1, 5184, 3969, 324, 2025, 1296, 81, 324, 81, 1, 8100, 1600, 4900, 225, 100, 400, 900, 25, 100, 1, 12100, 9801

Offset: 1

Author

N. J. A. Sloane, May 26 2001

Keywords

• nonn
• tabl
• easy
• nice
• frac

Comments

Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).

Examples

			Triangle 1/m^2-1/n^2, m >= 1, 1<=n<=m, (i.e. with rows reversed) begins
0
3/4, 0
8/9, 5/36, 0
15/16, 3/16, 7/144, 0
24/25, 21/100, 16/225, 9/400, 0
35/36, 2/9, 1/12, 5/144, 11/900, 0
		

References

• J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 77.

Links

• J. J. O'Connor and E. F. Robertson, Johannes Robert Rydberg
• Eric Weisstein's World of Physics, Balmer Formula
• Reinhard Zumkeller, Rows n=..100 of triangle, flattened

Crossrefs

Cf. A061035. Rows give A061037-A061050.

Cf. A126252.

Programs

Haskell

import Data.Ratio ((%), denominator)
a061036 n k = a061036_tabl !! (n-1) !! (k-1)
a061036_row = map denominator . balmer where
   balmer n = map (subtract (1 % n ^ 2) . (1 %) . (^ 2)) [n, n-1 .. 1]
a061036_tabl = map a061036_row [1..]
-- Reinhard Zumkeller, Apr 12 2012

Mathematica

t[m_, n_] := Denominator[1/m^2 - 1/n^2]; Table[t[m, n], {n, 1, 12}, {m, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 17 2012 *)

Extensions

More terms from Naohiro Nomoto, Jul 15 2001