A171522 Denominator of 1/n^2-1/(n+2)^2.

0, 9, 16, 225, 144, 1225, 576, 3969, 1600, 9801, 3600, 20449, 7056, 38025, 12544, 65025, 20736, 104329, 32400, 159201, 48400, 233289, 69696, 330625, 97344, 455625, 132496, 613089, 176400, 808201, 230400, 1046529, 295936, 1334025, 374544, 1677025, 467856

Offset: 0

Author

Paul Curtz, Dec 11 2009

Keywords

• nonn
• easy
• frac

Comments

This is the third column in the table of denominators of the hydrogenic spectra (the main diagonal A147560):

0, 0, 0, 0, 0, 0, 0, 0... A000004

1, 4, 9, 16, 25, 36, 49, 64... A000290

1, 36, 16, 100, 9, 196, 64, 324... A061038

1, 144, 225, 12, 441, 576, 81, 900... A061040

1, 400, 144, 784, 64,1296, 400,1936... A061042

1, 900 1225,1600,2025, 100,3025,3600... A061044

1,1764, 576, 324, 225,4356, 48,6084... A061046

1,3136,3969,4900,5929,7056,8281, 196... A061048.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Crossrefs

Cf. A105371. Bisections: A060300, A069075.

Programs

Maple

A171522 := proc(n) if n = 0 then 0 else lcm(n+2,n) ; %^2 ; end if ; end:
seq(A171522(n),n=0..70) ; # R. J. Mathar, Dec 15 2009

Mathematica

a[n_] := If[EvenQ[n], (n*(n+2))^2/4, (n*(n+2))^2]; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Jun 13 2017 *)

PARI

concat(0, Vec(x*(x^8+4*x^6+16*x^5+190*x^4+64*x^3+180*x^2+16*x+9) / ((x-1)^5*-(x+1)^5) + O(x^100))) \\ Colin Barker, Nov 05 2014

Formula

a(n) = (A066830(n+1))^2.

a(n) = -((-5+3*(-1)^n)*n^2*(2+n)^2)/8. - Colin Barker, Nov 05 2014

G.f.: x*(x^8+4*x^6+16*x^5+190*x^4+64*x^3+180*x^2+16*x+9) / ((x-1)^5*-(x+1)^5). - Colin Barker, Nov 05 2014

Extensions

Edited and extended by R. J. Mathar, Dec 15 2009