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Reduced denominators of (n-1)^2/(n^2 + n + 1).
6

%I #29 Oct 29 2022 04:55:30

%S 1,7,13,7,31,43,19,73,91,37,133,157,61,211,241,91,307,343,127,421,463,

%T 169,553,601,217,703,757,271,871,931,331,1057,1123,397,1261,1333,469,

%U 1483,1561,547,1723,1807,631,1981,2071,721,2257,2353,817,2551,2653

%N Reduced denominators of (n-1)^2/(n^2 + n + 1).

%C Arises in Routh's theorem.

%H G. C. Greubel, <a href="/A046163/b046163.txt">Table of n, a(n) for n = 1..5000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RouthsTheorem.html">Routh's Theorem</a>.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,3,0,0,-3,0,0,1).

%F G.f.: x*(1 + 7*x + 13*x^2 + 4*x^3 + 10*x^4 + 4*x^5 + x^6 + x^7 + x^8)/(1 - x^3)^3.

%F From _Amiram Eldar_, Aug 11 2022: (Start)

%F a(n) = numerator((n^2 + n + 1)/3).

%F Sum_{n>=1} 1/a(n) = (2*tanh(Pi/(2*sqrt(3))) + 3*tanh(sqrt(3)*Pi/2))*Pi/(3*sqrt(3)) - 1. (End)

%t a[n_] := Denominator[(n - 1)^2/(n^2 + n + 1)]; Array[a, 50] (* _Amiram Eldar_, Aug 11 2022 *)

%o (Magma) [Denominator((n-1)^2/(n^2+n+1)): n in [1..70]]; // _G. C. Greubel_, Oct 27 2022

%o (SageMath) [denominator((n-1)^2/(n^2+n+1)) for n in range(1,71)] # _G. C. Greubel_, Oct 27 2022

%Y Cf. A046162 (numerators).

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_