A129203 a(n) = numerator(3/(n+1)^3)*(3/2 + (-1)^n/2).

6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1

Offset: 0

Author

Paul Barry, Apr 03 2007

Keywords

• nonn
• easy

Comments

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

Periodic with period 6. - Alois P. Heinz, Oct 03 2012

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Crossrefs

Cf. sequences listed in Comments section of A283393.

Programs

GAP

List(List([0..10],n->3/(n+1)^3*(3/2+(-1)^n/2)),NumeratorRat); # Muniru A Asiru, Jul 01 2018

Magma

[Numerator(3/(n+1)^3)*(3/2 + (-1)^n/2): n in [1..100]]; // Vincenzo Librandi, Jul 01 2018

Magma

&cat [[6, 3, 2, 3, 6, 1]^^20]; // Vincenzo Librandi, Jul 01 2018

Maple

a:= n-> [6, 3, 2, 3, 6, 1][irem(n, 6)+1]:
seq(a(n), n=0..119);  # Alois P. Heinz, Oct 03 2012

Mathematica

Array[Numerator[3/(# + 1)^3] (3/2 + (-1)^#/2) &, 105, 0] (* or *)
PadRight[{}, 105, {6, 3, 2, 3, 6, 1}] (* Michael De Vlieger, Jun 30 2018 *)
CoefficientList[ Series[-(x^5 + 6x^4 + 3x^3 + 2x^2 + 3x + 6)/(x^6 - 1), {x, 0,
   105}], x] (* or *)
LinearRecurrence[{0, 0, 0, 0, 0, 1}, {6, 3, 2, 3, 6, 1}, 105] (* Robert G. Wilson v, Jul 28 2018 *)

PARI

a(n)=[6, 3, 2, 3, 6, 1][n%6+1] \\ Charles R Greathouse IV, Oct 28 2014

Sage

def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
def A129203(n): return Gauss_factorial(3, n+1)
[A129203(j) for j in (0..71)] # Peter Luschny, Oct 01 2012

Formula

G.f.: (6 + 3*x + 2*x^2 + 3*x^3 + 6*x^4 + x^5)/(1 - x^6).

a(n) = cos(2*Pi*n/3) + sqrt(3)*sin(2*Pi*n/3) + cos(Pi*n/3)/3 - sqrt(3)*sin(Pi*n/3)/3 + 7*cos(Pi*n)/6 + 7/2.

a(n) = numerator(6/(n+1)^2). - Paul Barry, Apr 03 2007

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

a(n) = 3_(n+1)! = Gauss_factorial(3, n+1) = Product_{1<=j<=3, gcd(j,n+1)=1} j. - Peter Luschny, Oct 01 2012

a(n) = denominator((n+1)/6). - Jon Hearn, Nov 10 2013

a(n) = denominator of 2*(1/(12*n)+1)*n^n; related to gamma function approximation for positive integers less the factor sqrt(Pi/2)/(exp(n)*sqrt(n)). - Thomas Blankenhorn, Jun 21 2018

a(n) = 6/gcd(n+1,6). - Ridouane Oudra, Jul 29 2022