A046163 Reduced denominators of (n-1)^2/(n^2 + n + 1).

1, 7, 13, 7, 31, 43, 19, 73, 91, 37, 133, 157, 61, 211, 241, 91, 307, 343, 127, 421, 463, 169, 553, 601, 217, 703, 757, 271, 871, 931, 331, 1057, 1123, 397, 1261, 1333, 469, 1483, 1561, 547, 1723, 1807, 631, 1981, 2071, 721, 2257, 2353, 817, 2551, 2653

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

Arises in Routh's theorem.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Eric Weisstein's World of Mathematics, Routh's Theorem.
• Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Crossrefs

Cf. A046162 (numerators).

Programs

Magma

[Denominator((n-1)^2/(n^2+n+1)): n in [1..70]]; // G. C. Greubel, Oct 27 2022

Mathematica

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

SageMath

[denominator((n-1)^2/(n^2+n+1)) for n in range(1,71)] # G. C. Greubel, Oct 27 2022

Formula

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.

From Amiram Eldar, Aug 11 2022: (Start)

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

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