|
%I #15 Jun 13 2017 07:44:18
%S 0,9,16,225,144,1225,576,3969,1600,9801,3600,20449,7056,38025,12544,
%T 65025,20736,104329,32400,159201,48400,233289,69696,330625,97344,
%U 455625,132496,613089,176400,808201,230400,1046529,295936,1334025,374544,1677025,467856
%N Denominator of 1/n^2-1/(n+2)^2.
%C This is the third column in the table of denominators of the hydrogenic spectra (the main diagonal A147560):
%C 0, 0, 0, 0, 0, 0, 0, 0... A000004
%C 1, 4, 9, 16, 25, 36, 49, 64... A000290
%C 1, 36, 16, 100, 9, 196, 64, 324... A061038
%C 1, 144, 225, 12, 441, 576, 81, 900... A061040
%C 1, 400, 144, 784, 64,1296, 400,1936... A061042
%C 1, 900 1225,1600,2025, 100,3025,3600... A061044
%C 1,1764, 576, 324, 225,4356, 48,6084... A061046
%C 1,3136,3969,4900,5929,7056,8281, 196... A061048.
%H Colin Barker, <a href="/A171522/b171522.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,0,-10,0,10,0,-5,0,1).
%F a(n) = (A066830(n+1))^2.
%F a(n) = -((-5+3*(-1)^n)*n^2*(2+n)^2)/8. - _Colin Barker_, Nov 05 2014
%F 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
%p A171522 := proc(n) if n = 0 then 0 else lcm(n+2,n) ; %^2 ; end if ; end:
%p seq(A171522(n),n=0..70) ; # _R. J. Mathar_, Dec 15 2009
%t 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 *)
%o (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
%Y Cf. A105371. Bisections: A060300, A069075.
%K nonn,easy,frac
%O 0,2
%A _Paul Curtz_, Dec 11 2009
%E Edited and extended by _R. J. Mathar_, Dec 15 2009
|
