A061036 -id:A061036 - cp's OEIS frontend

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

N. J. A. Sloane, May 26 2001

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

			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

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

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

Cf. A061036. Rows give A061037-A061050.

Cf. A126252.

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

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 *)

More terms from Naohiro Nomoto, Jul 15 2001

A174413 Triangle T(n,m) with the denominator of 1/(n-m)^2-1/n^2, read by rows, 1<=m

Original entry on oeis.org

4, 36, 9, 144, 16, 16, 400, 225, 100, 25, 900, 144, 12, 9, 36, 1764, 1225, 784, 441, 196, 49, 3136, 576, 1600, 64, 576, 64, 64, 5184, 3969, 324, 2025, 1296, 81, 324, 81, 8100, 1600, 4900, 225, 100, 400, 900, 25, 100, 12100, 9801, 7744, 5929, 4356, 3025, 1936, 1089, 484, 121

Offset: 2

Paul Curtz, Mar 19 2010

Obtained by deleting the last entry in each row of A061036 or by reversing rows in A120073.

			Triangle T(n,m) begins:
     4,
    36,    9,
   144,   16,   16,
   400,  225,  100,  25,
   900,  144,   12,   9,  36,
  1764, 1225,  784, 441, 196, 49,
  3136,  576, 1600,  64, 576, 64, 64,

Alois P. Heinz, Table of n, a(n) for n = 2..2017

Cf. A165441, A172370 (numerators).

A174413 := proc(n,m) 1/(n-m)^2-1/n^2 ; denom(%) ; end proc:
seq(seq(A174413(n, k), k=1..n-1), n=2..11); # R. J. Mathar, Jan 27 2011

T[n_, m_] := Denominator[1/(n - m)^2 - 1/n^2];
Table[T[n, m], {n, 2, 11}, {m, 1, n-1}] // Flatten (* Jean-François Alcover, May 18 2018 *)

cp's OEIS Frontend

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

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