A171638 Denominator of 1/(n-2)^2 - 1/(n+2)^2.

0, 25, 9, 441, 64, 2025, 225, 5929, 576, 13689, 1225, 27225, 2304, 48841, 3969, 81225, 6400, 127449, 9801, 190969, 14400, 275625, 20449, 385641, 28224, 525625, 38025, 700569, 50176, 915849, 65025, 1177225, 82944, 1490841

Offset: 2

Author

Paul Curtz, Dec 13 2009

Keywords

• nonn
• frac
• easy

Comments

Fifth column of an array of denominators related to the energies of the hydrogen spectrum, mentioned in A171522. At n=2, the defining formula has a pole and is replaced by 0 to conform with A171621 and A099761.

Links

• G. C. Greubel, Table of n, a(n) for n = 2..5000
• Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Crossrefs

Cf. A099761, A171621.

Programs

Magma

[0] cat [Denominator((1/(n-2)^2 -1/(n+2)^2)): n in [3..350]]; // Bruno Berselli, Apr 05 2011

Maple

A061037 := proc(n) 1/4-1/n^2 ; numer(%) ; end proc:
A171621 := proc(n) if n mod 4 = 2 then 4*A061037(n) ; else A061037(n) ; end if; end proc:
A171638 := proc(n) A171621(n)^2 ; end proc:
seq(A171638(n),n=2..90) ; # R. J. Mathar, Apr 02 2011

Mathematica

Table[If[n == 2, 0, Denominator[1/(n-2)^2 - 1/(n+2)^2]], {n, 2, 50}] (* G. C. Greubel, Sep 20 2018 *)
LinearRecurrence[{0,5,0,-10,0,10,0,-5,0,1},{0,25,9,441,64,2025,225,5929,576,13689},50] (* Harvey P. Dale, Sep 07 2021 *)

PARI

for(n=2,100, print1(if(n==2,0, denominator(1/(n-2)^2 - 1/(n+2)^2)), ", ")) \\ G. C. Greubel, Sep 20 2018

Formula

a(n) = (A171621(n))^2.

a(2*n+2) = A099761(n).

G.f.: -((x(25+9*x+316*x^2+19*x^3+70*x^4-5*x^5-36*x^6+x^7+9*x^8))/((-1+x)^5 (1+x)^5)). - Harvey P. Dale, Sep 07 2021

Sum_{n>=3} 1/a(n) = 19*Pi^2/192 - 115/144. - Amiram Eldar, Aug 14 2022