A165441 Table T(k,n) read by antidiagonals: denominator of 1/min(n,k)^2 -1/max(n,k)^2.

1, 4, 4, 9, 1, 9, 16, 36, 36, 16, 25, 16, 1, 16, 25, 36, 100, 144, 144, 100, 36, 49, 9, 225, 1, 225, 9, 49, 64, 196, 12, 400, 400, 12, 196, 64, 81, 64, 441, 144, 1, 144, 441, 64, 81, 100, 324, 576, 784, 900, 900, 784, 576, 324, 100, 121, 25, 81, 64, 1225, 1, 1225, 64, 81, 25, 121

Offset: 1

Author

Paul Curtz, Sep 19 2009

Keywords

• nonn
• tabl
• frac
• look
• easy

Comments

A synopsis of the denominators of the transitions in the Rydberg-Ritz spectrum of hydrogenic atoms.

Examples

			.1,   4,   9,   16,   25,   36,   49,   64,   81, ... A000290
.4,   1,  36,   16,  100,    9,  196,   64,  324, ... A061038
.9,  36,   1,  144,  225,   12,  441,  576,   81, ... A061040
16,  16, 144,    1,  400,  144,  784,   64, 1296, ... A061042
25, 100, 225,  400,    1,  900, 1225, 1600, 2025, ... A061044
36,   9,  12,  144,  900,    1, 1764,  576,  324, ... A061046
49, 196, 441,  784, 1225, 1764,    1, 3136, 3969, ... A061048
64,  64, 576,   64, 1600,  576, 3136,    1, 5184, ... A061050
81, 324,  81, 1296, 2025,  324, 3969, 5184,    1, ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1035

Programs

Maple

T:= (k,n)-> denom(1/min (n,k)^2 -1/max (n, k)^2):
seq(seq(T(k, d-k), k=1..d-1), d=2..12);

Mathematica

T[n_, k_] := Denominator[1/Min[n, k]^2 - 1/Max[n, k]^2];
Table[T[n-k, k], {n, 2, 12}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 04 2020 *)

Formula

T(n,k) = A165727(n,k).

Extensions

Edited by R. J. Mathar, Feb 27 2010, Mar 03 2010