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

0,2

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A105371. Bisections: A060300, A069075.

Sequence in context: A075373 A072470 A053911 * A236287 A329964 A050802

Adjacent sequences: A171519 A171520 A171521 * A171523 A171524 A171525

Keyword

nonn,easy,frac

Author

Paul Curtz, Dec 11 2009

Extensions

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

Status

approved