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

2,1

Comments

The corresponding numerator triangle is A120072.

See A120072 and A120070 for more details.

Links

G. C. Greubel, Rows n = 2..50 of the triangle, flattened

Wofdieter Lang, First ten rows, rationals and more.

Formula

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

Example

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;

Mathematica

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

Prog

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

Crossrefs

Row sums give A120075.

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

Sequence in context: A134815 A367992 A352002 * A056894 A272145 A272221

Adjacent sequences: A120070 A120071 A120072 * A120074 A120075 A120076

Keyword

nonn,easy,tabl,frac

Author

Wolfdieter Lang, Jul 20 2006

Status

approved