A099270 - OEIS

%I #19 Jul 02 2023 18:55:04

%S 0,1,12,169,2352,32761,456300,6355441,88519872,1232922769,17172398892,

%T 239180661721,3331356865200,46399815451081,646266059449932,

%U 9001325016847969,125372284176421632,1746210653453054881

%N Unsigned member r=-12 of the family of Chebyshev sequences S_r(n) defined in A092184.

%C ((-1)^(n+1))*a(n) = S_{-12}(n), n>=0, defined in A092184.

%H Vincenzo Librandi, <a href="/A099270/b099270.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (13, 13, -1).

%F a(n) = (T(n, 7)-(-1)^n)/8, with Chebyshev's polynomials of the first kind evaluated at x=7: T(n, 7)=A011943(n)=((7+4*sqrt(3))^n + (7-4*sqrt(3))^n)/2.

%F a(n) = 13*a(n-1) + 13*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=12.

%F G.f.: x*(1-x)/((1+x)*(1-14*x+x^2)) = x*(1-x)/(1-13*x-13*x^2+x^3) (from the Stephan link, see A092184).

%F a(n) = 14*a(n-1)-a(n-2)-2*(-1)^n, a(0)=0, a(1)=1. a(-n)=a(n).

%t a[n_] := (ChebyshevT[n, 7] - (-1)^n)/8; Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Jun 21 2013, from 1st formula *)

%t CoefficientList[Series[x (1 - x) / ((1 + x) (1 - 14 x + x^2)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jun 21 2013 *)

%o (PARI) a(n)=real(((7+4*quadgen(12))^n-(-1)^n)/8) /* _Michael Somos_, Apr 30 2005 */

%o (PARI) a(n)=n=abs(2*n); round(2^(n-4)*prod(k=1,n,2-sin(2*Pi*k/n)))

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, Oct 18 2004