A129196 a(n) = denominator(3*(3+(-1)^n)/(n+1)^3).

1, 4, 9, 32, 125, 36, 343, 256, 243, 500, 1331, 288, 2197, 1372, 1125, 2048, 4913, 972, 6859, 4000, 3087, 5324, 12167, 2304, 15625, 8788, 6561, 10976, 24389, 4500, 29791, 16384, 11979, 19652, 42875, 7776, 50653, 27436, 19773, 32000, 68921, 12348, 79507, 42592

Offset: 0

Author

Paul Barry, Apr 02 2007, Apr 03 2007

Keywords

• nonn
• frac
• easy

Comments

Numerator of 3*(3+(-1)^n)/(n+1)^3 is A129197.

(1/(2*Pi))*Integral_{t=0..2*Pi} exp(i*(n+1)*t)*((t-Pi)/i)^3 dt = (A129202(n)*Pi^2-A129203(n))/A129196(n) with i=sqrt(-1).

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,4,0,0,0,0,0,-6,0,0,0,0,0,4,0,0,0,0,0,-1).

Crossrefs

Cf. A129196, A129197, A129202, A129203, A129204.

Programs

Mathematica

a[n_] := Denominator[3*(3 + (-1)^n)/(n + 1)^3]; Array[a, 50, 0] (* Amiram Eldar, Sep 11 2022 *)

Formula

a(n) = A129204(n+1)/(5/3+(4/3)*cos(2*Pi*(n+1)/3)).

a(n) = denominator((1/(2*Pi))*Integral_{t=0..2*Pi} exp(i*(n+1)*t)*((t-Pi)/i)^3 dt) with i=sqrt(-1).

a(n) = denominator((Pi^2*(n+1)^2-6)/(n+1)^3).

a(n) = ((n+1)^3/(gcd(n+1,2)*gcd(n+1,3))). - Paul Barry, Oct 09 2007

a(n) = numerator of coefficient of x^6 in the Maclaurin expansion of -exp(-(n+1)*x^2). - Francesco Daddi, Aug 04 2011

Sum_{n>=0} 1/a(n) = 29*zeta(3)/24. - Amiram Eldar, Sep 11 2022