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
From Amiram Eldar, Feb 03 2026: (Start)
Sum_{n>=1} 1/a(n) = 7*Pi^2/48 - 5/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/4 - Pi^2/48. (End)
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
KEYWORD
nonn,easy,frac
AUTHOR
Paul Curtz, Dec 11 2009
EXTENSIONS
Edited and extended by R. J. Mathar, Dec 15 2009
STATUS
approved