A129202 Denominator of 3*(3+(-1)^n) / (n+1)^2.
1, 2, 3, 8, 25, 6, 49, 32, 27, 50, 121, 24, 169, 98, 75, 128, 289, 54, 361, 200, 147, 242, 529, 96, 625, 338, 243, 392, 841, 150, 961, 512, 363, 578, 1225, 216, 1369, 722, 507, 800, 1681, 294, 1849, 968, 675, 1058, 2209, 384, 2401, 1250, 867, 1352, 2809, 486
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Peter Bala, A note on the sequence of numerators of a rational function , Feb 2019.
- Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,3,0,0,0,0,0,-3,0,0,0,0,0,1).
Programs
-
Magma
[Denominator(3*(3+(-1)^n)/(n+1)^2): n in [0..50]]; // G. C. Greubel, Oct 26 2017
-
Maple
A129202:=n->numer((n+1)/2)*numer((n+1)/3): seq(A129202(n), n=0..100); # Wesley Ivan Hurt, Jul 18 2014
-
Mathematica
Table[Numerator[(n + 1)/2] Numerator[(n + 1)/3], {n, 0, 100}] (* Wesley Ivan Hurt, Jul 18 2014 *) LinearRecurrence[{0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 1}, {1, 2, 3, 8, 25, 6, 49, 32, 27, 50, 121, 24, 169, 98, 75, 128, 289, 54}, 60] (* Harvey P. Dale, Nov 20 2016 *) -
PARI
for(n=0,50, print1(denominator(3*(3+(-1)^n)/(n+1)^2), ", ")) \\ G. C. Greubel, Oct 26 2017
Formula
a(n) = A129196(n)/(n+1).
(1/(2*Pi))*Integral_{t=0..2*Pi} exp(i*(n+1)*t)*((t-Pi)/i)^3 dt = (a(n)*Pi^2-A129203(n))/A129196(n), i=sqrt(-1).
a(n) = ( Numerator of (n+1)/2 ) * ( Numerator of (n+1)/3 ) = A026741(n+1) * A051176(n+1). - Wesley Ivan Hurt, Jul 18 2014
G.f.: -(x^16 +2*x^15 +3*x^14 +8*x^13 +25*x^12 +6*x^11 +46*x^10 +26*x^9 +18*x^8 +26*x^7 +46*x^6 +6*x^5 +25*x^4 +8*x^3 +3*x^2 +2*x +1) / ((x -1)^3*(x +1)^3*(x^2 -x +1)^3*(x^2 +x +1)^3). - Colin Barker, Jul 18 2014
a(n+18) = 3*a(n+12)-3*a(n+6)+a(n). - Robert Israel, Jul 18 2014
a(n) = 2*(n+1)^2 * (7-4*cos(2*Pi*(n+1)/3)) / (9*(3-(-1)^n)). - Vaclav Kotesovec, Jul 20 2014
From Peter Bala, Feb 27 2019: (Start)
The following remarks assume an offset of 1.
a(n) = n^2/gcd(n,6) = n*A060789(n).
a(n) = n^2/b(n), where b(n) is the purely periodic sequence [1,2,3,2,1,6,...] with period 6. Thus a(n) is a quasi-polynomial in n:
a(6*n+1) = (6*n + 1)^2;
a(6*n+2) = 2*(3*n + 1)^2;
a(6*n+3) = 3*(2*n + 1)^2;
a(6*n+4) = 2*(3*n + 2)^2;
a(6*n+5) = (6*n + 5)^2;
a(6*n) = 6*n^2.
O.g.f.: F(x) - 2*F(x^2) - 6*F(x^3) + 12*F(x^6), where F(x) = x*(1 + x)/(1 - x)^3 is the generating function for the squares. (End)
Sum_{n>=0} 1/a(n) = 55*Pi^2/216. - Amiram Eldar, Sep 27 2022
Extensions
More terms from Wesley Ivan Hurt, Jul 18 2014
Comments