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%I #42 Dec 30 2022 05:30:16
%S 0,1,4,3,16,25,12,49,64,27,100,121,48,169,196,75,256,289,108,361,400,
%T 147,484,529,192,625,676,243,784,841,300,961,1024,363,1156,1225,432,
%U 1369,1444,507,1600,1681,588,1849,1936,675,2116,2209,768,2401,2500
%N Reduced numerators of (n-1)^2/(n^2 + n + 1).
%C Arises in Routh's theorem.
%C With offset 0, multiplicative with a(3^e) = 3^(2e-1), a(p^e) = p^(2e) otherwise. - _David W. Wilson_, Jun 12 2005, corrected by _Robert Israel_, Apr 28 2017
%H Robert Israel, <a href="/A046162/b046162.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RouthsTheorem.html">Routh's Theorem</a>.
%F G.f.: x^2*(1 + 4*x + 3*x^2 + 13*x^3 + 13*x^4 + 3*x^5 + 4*x^6 + x^7)/(1 - x^3)^3.
%F a(n) = (n-1)^2/3 if n-1 == 0 (mod 3), (n-1)^2 otherwise. - _David W. Wilson_, Jun 12 2005, corrected by _Robert Israel_, Apr 28 2017
%F From _Amiram Eldar_, Aug 11 2022: (Start)
%F a(n) = numerator((n-1)^2/3).
%F Sum_{n>=2} 1/a(n) = 11*Pi^2/54. (End)
%F From _Amiram Eldar_, Dec 30 2022: (Start)
%F With offset 0, Dirichlet g.f.: zeta(s-2)*(1-6/3^s).
%F Sum_{k=1..n} a(k) ~ 7*n^3/27. (End)
%p seq(numer((n-1)^2/(n^2+n+1)), n=1..51) ; # _Zerinvary Lajos_, Jun 04 2008
%p seq(denom(3/n^2-2), n=0..76) ; # _Zerinvary Lajos_, Jun 04 2008
%t a[n_] := Numerator[(n - 1)^2/(n^2 + n + 1)]; Array[a, 50] (* _Amiram Eldar_, Aug 11 2022 *)
%o (Magma) [Numerator((n-1)^2/3): n in [1..70]]; // _G. C. Greubel_, Oct 27 2022
%o (SageMath) [numerator((n-1)^2/3) for n in range(1,71)] # _G. C. Greubel_, Oct 27 2022
%Y Cf. A046163 (denominators), A147560.
%K nonn,mult
%O 1,3
%A _Eric W. Weisstein_
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