A046162 Reduced numerators of (n-1)^2/(n^2 + n + 1).

0, 1, 4, 3, 16, 25, 12, 49, 64, 27, 100, 121, 48, 169, 196, 75, 256, 289, 108, 361, 400, 147, 484, 529, 192, 625, 676, 243, 784, 841, 300, 961, 1024, 363, 1156, 1225, 432, 1369, 1444, 507, 1600, 1681, 588, 1849, 1936, 675, 2116, 2209, 768, 2401, 2500

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• mult

Comments

Arises in Routh's theorem.

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Routh's Theorem.

Crossrefs

Cf. A046163 (denominators), A147560.

Programs

Magma

[Numerator((n-1)^2/3): n in [1..70]]; // G. C. Greubel, Oct 27 2022

Maple

seq(numer((n-1)^2/(n^2+n+1)), n=1..51) ; # Zerinvary Lajos, Jun 04 2008
seq(denom(3/n^2-2), n=0..76) ; # Zerinvary Lajos, Jun 04 2008

Mathematica

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

SageMath

[numerator((n-1)^2/3) for n in range(1,71)] # G. C. Greubel, Oct 27 2022

Formula

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.

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

From Amiram Eldar, Aug 11 2022: (Start)

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

Sum_{n>=2} 1/a(n) = 11*Pi^2/54. (End)

From Amiram Eldar, Dec 30 2022: (Start)

With offset 0, Dirichlet g.f.: zeta(s-2)*(1-6/3^s).

Sum_{k=1..n} a(k) ~ 7*n^3/27. (End)