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Denominator of 3*(3+(-1)^n) / (n+1)^2.
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%I #42 Sep 20 2024 21:34:54

%S 1,2,3,8,25,6,49,32,27,50,121,24,169,98,75,128,289,54,361,200,147,242,

%T 529,96,625,338,243,392,841,150,961,512,363,578,1225,216,1369,722,507,

%U 800,1681,294,1849,968,675,1058,2209,384,2401,1250,867,1352,2809,486

%N Denominator of 3*(3+(-1)^n) / (n+1)^2.

%C A divisibility sequence, that is, if n divides m then a(n) divides a(m). - _Peter Bala_, Feb 27 2019

%H G. C. Greubel, <a href="/A129202/b129202.txt">Table of n, a(n) for n = 0..1000</a>

%H Peter Bala, <a href="/A306367/a306367.pdf">A note on the sequence of numerators of a rational function </a>, Feb 2019.

%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,3,0,0,0,0,0,-3,0,0,0,0,0,1).

%F a(n) = A129196(n)/(n+1).

%F (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).

%F a(n) = ( Numerator of (n+1)/2 ) * ( Numerator of (n+1)/3 ) = A026741(n+1) * A051176(n+1). - _Wesley Ivan Hurt_, Jul 18 2014

%F 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

%F a(n+18) = 3*a(n+12)-3*a(n+6)+a(n). - _Robert Israel_, Jul 18 2014

%F a(n) = 2*(n+1)^2 * (7-4*cos(2*Pi*(n+1)/3)) / (9*(3-(-1)^n)). - _Vaclav Kotesovec_, Jul 20 2014

%F From _Peter Bala_, Feb 27 2019: (Start)

%F The following remarks assume an offset of 1.

%F a(n) = n^2/gcd(n,6) = n*A060789(n).

%F 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:

%F a(6*n+1) = (6*n + 1)^2;

%F a(6*n+2) = 2*(3*n + 1)^2;

%F a(6*n+3) = 3*(2*n + 1)^2;

%F a(6*n+4) = 2*(3*n + 2)^2;

%F a(6*n+5) = (6*n + 5)^2;

%F a(6*n) = 6*n^2.

%F 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)

%F Sum_{n>=0} 1/a(n) = 55*Pi^2/216. - _Amiram Eldar_, Sep 27 2022

%p A129202:=n->numer((n+1)/2)*numer((n+1)/3): seq(A129202(n), n=0..100); # _Wesley Ivan Hurt_, Jul 18 2014

%t Table[Numerator[(n + 1)/2] Numerator[(n + 1)/3], {n, 0, 100}] (* _Wesley Ivan Hurt_, Jul 18 2014 *)

%t 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 *)

%o (PARI) for(n=0,50, print1(denominator(3*(3+(-1)^n)/(n+1)^2), ", ")) \\ _G. C. Greubel_, Oct 26 2017

%o (Magma) [Denominator(3*(3+(-1)^n)/(n+1)^2): n in [0..50]]; // _G. C. Greubel_, Oct 26 2017

%Y Cf. A026741, A051176, A129196, A129197 (numerators), A060789.

%K nonn,frac,easy

%O 0,2

%A _Paul Barry_, Apr 03 2007

%E More terms from _Wesley Ivan Hurt_, Jul 18 2014