A099270 Unsigned member r=-12 of the family of Chebyshev sequences S_r(n) defined in A092184.

0, 1, 12, 169, 2352, 32761, 456300, 6355441, 88519872, 1232922769, 17172398892, 239180661721, 3331356865200, 46399815451081, 646266059449932, 9001325016847969, 125372284176421632, 1746210653453054881

Offset

0,3

Comments

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for two-way infinite sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (13, 13, -1).

Formula

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.

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

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).

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

Mathematica

a[n_] := (ChebyshevT[n, 7] - (-1)^n)/8; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jun 21 2013, from 1st formula *)
CoefficientList[Series[x (1 - x) / ((1 + x) (1 - 14 x + x^2)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 21 2013 *)

Prog

(PARI) a(n)=real(((7+4*quadgen(12))^n-(-1)^n)/8) /* Michael Somos, Apr 30 2005 */
(PARI) a(n)=n=abs(2*n); round(2^(n-4)*prod(k=1, n, 2-sin(2*Pi*k/n)))

Crossrefs

Sequence in context: A071103 A012489 A027772 * A187361 A366235 A239335

Adjacent sequences: A099267 A099268 A099269 * A099271 A099272 A099273

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 18 2004

Status

approved