A171372 a(n) = Numerator of 1/(2*n)^2 - 1/(3*n)^2 for n > 0, a(0) = 1.

1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5

Offset: 0

Author

Paul Curtz, Dec 07 2009

Keywords

• nonn
• easy
• frac

Comments

The diagonal of a table of numerators of the Rydberg-Ritz spectrum of hydrogen:

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012

0, 5, 3, 21, 2, 45, 15, 77, 6, 117, 35, ... A061037

0, 9, 5, 33, 3, 65, 21, 105, 1, 153, 45, ... A061041

0, 13, 7, 5, 4, 85, 1, 133, 10, 7, 55, ... A061045

0, 17, 9, 57, 5, 105, 33, 161, 3, 225, 65, ... A061049

0, 21, 11, 69, 6, 1, 39, 189, 14, 261, 3, ...

0, 25, 13, 1, 7, 145, 5, 217, 1, 11, 85, ...

0, 29, 15, 93, 8, 165, 51, 5, 18, 333, 95, ...

0, 33, 17, 105, 9, 185, 57, 273, 5, 369, 105, ...

0, 37, 19, 13, 10, 205, 7, 301, 22, 5, 115, ...

0, 41, 21, 129, 11, 9, 69, 329, 3, 441, 1, ...

In that respect, constructed similar to A144437.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Crossrefs

Cf. A171373 (binomial transform), A171408, A105371.

Programs

Magma

[1] cat [Numerator(5/(6*n)^2): n in [1..100]]; // G. C. Greubel, Sep 20 2018

Mathematica

Table[If[n==0,1,Numerator[5/(6*n)^2]], {n,0,100}] (* G. C. Greubel, Sep 20 2018 *)

PARI

concat([1], vector(100, n, numerator(5/(6*n)^2))) \\ G. C. Greubel, Sep 20 2018

Formula

a(n) = numerator of 5/(6*n)^2 .

Period 5: repeat [1,5,5,5,5].

G.f.: (1 + 5*x + 5*x^2 + 5*x^3 + 5*x^4)/((1-x)*(1 + x + x^2 + x^3 + x^4)).

a(n) = 1 + 4*sign(n mod 5). - Wesley Ivan Hurt, Sep 26 2018

a(n) = (21-8*cos(2*n*Pi/5)-8*cos(4*n*Pi/5))/5. - Wesley Ivan Hurt, Sep 27 2018

Extensions

Edited by R. J. Mathar, Dec 15 2009