A061035 Triangle T(m,n) = numerator of 1/m^2 - 1/n^2, n >= 1, m=n,n-1,n-2,...,1.

0, 0, 3, 0, 5, 8, 0, 7, 3, 15, 0, 9, 16, 21, 24, 0, 11, 5, 1, 2, 35, 0, 13, 24, 33, 40, 45, 48, 0, 15, 7, 39, 3, 55, 15, 63, 0, 17, 32, 5, 56, 65, 8, 77, 80, 0, 19, 9, 51, 4, 3, 21, 91, 6, 99, 0, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 0, 23, 11, 7, 5, 95, 1, 119, 1, 5, 35, 143, 0

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

• Reinhard Zumkeller, Rows n=1..100 of triangle, flattened
• J. J. O'Connor and E. F. Robertson, Johannes Robert Rydberg
• Eric Weisstein's World of Physics, Balmer Formula

Crossrefs

Cf. A061036. Rows give A061037-A061050.

Cf. A126252.

Programs

Haskell

import Data.Ratio ((%), numerator)
a061035 n k = a061035_tabl !! (n-1) !! (k-1)
a061035_row = map numerator . balmer where
   balmer n = map (subtract (1 % n ^ 2) . (1 %) . (^ 2)) [n, n-1 .. 1]
a061035_tabl = map a061035_row [1..]
-- Reinhard Zumkeller, Apr 12 2012

Mathematica

t[m_, n_] := Numerator[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