A120073 Denominator triangle for hydrogen spectrum rationals.

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

Offset: 2

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• tabl
• frac

Comments

The corresponding numerator triangle is A120072.

See A120072 and A120070 for more details.

Examples

			For the rational triangle see W. Lang link.
Denominator triangle begins as:
    4;
    9,  36;
   16,  16, 144;
   25, 100, 225,  400;
   36,   9,  12,  144,  900;
   49, 196, 441,  784, 1225, 1764;
   64,  64, 576,   64, 1600,  576, 3136;
   81, 324,  81, 1296, 2025,  324, 3969, 5184;
  100,  25, 900,  400,  100,  225, 4900, 1600, 8100;
		

Links

• G. C. Greubel, Rows n = 2..50 of the triangle, flattened
• Wofdieter Lang, First ten rows, rationals and more.

Crossrefs

Row sums give A120075.

Cf. A120070, A120072, A120074, A120076, A120077, A126252.

Programs

Magma

[Denominator(1/k^2 - 1/n^2): k in [1..n-1], n in [2..18]]; // G. C. Greubel, Apr 24 2023

Mathematica

Table[(1/n^2 - 1/m^2)//Denominator, {m,2,15}, {n,m-1}]//Flatten (* Jean-François Alcover, Sep 16 2013 *)

SageMath

def A120073(n,k): return denominator(1/k^2 - 1/n^2)
flatten([[A120073(n,k) for k in range(1,n)] for n in range(2,19)]) # G. C. Greubel, Apr 24 2023

Formula

a(m,n) = denominator(r(m,n)) with r(m,n) = 1/n^2 - 1/m^2, m>=2, n=1..m-1.