OFFSET
0,2
COMMENTS
Reciprocal Chebyshev polynomial of second kind evaluated at 3 multiplied by (-1)^n.
From Sharon Sela (sharonsela(AT)hotmail.com), Jan 19 2002: (Start)
a(n) is (-1)^n times the determinant of the following tridiagonal n X n matrix:
[2 3 0 . . . . . . .]
[3 2 3 0 . . . . . .]
[0 3 2 3 0 . . . . .]
[. 0 3 2 3 0 . . . .]
[. . . . . . . . . .]
[. . . . . . . . . .]
[. . . . 0 3 2 3 0 .]
[. . . . . 0 3 2 3 0]
[. . . . . . 0 3 2 3]
[. . . . . . . 0 3 2]
(End)
LINKS
FORMULA
a(n) = 3^n * ChebyshevU(n, -1/3).
a(n) = ( A088137(n+1) )^2 + ( A087455(n+1)/2 )^2 - ( A087455(n+2)/2 )^2. - Creighton Dement, Aug 20 2004
a(n) = -(2*a(n-1) + 9*a(n-2)) for n>1, with a(0)=1, a(1)=-2. - Philippe Deléham, Sep 19 2009
a(n) = (-2)^n*Product_{k=1..n}(1 + 3*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019
From G. C. Greubel, Jan 02 2024: (Start)
a(n) = (-1)^n * A127357(n).
E.g.f.: (1/4)*exp(-x)*(4*cos(2*sqrt(2)*x) - sqrt(2)*sin(2*sqrt(2)*x)). (End)
MATHEMATICA
Table[3^n ChebyshevU[n, -1/3], {n, 0, 24}]
PROG
(PARI) a(n)=if(n<0, 0, polcoeff(1/(1+2*x+9*x^2)+x*O(x^n), n))
(PARI) a(n)=if(n<0, 0, 3^n*subst(poltchebi(n+1)+3*poltchebi(n), 'x, -1/3)*3/8) /* Michael Somos, Sep 15 2005 */
(PARI) a(n)=if(n<0, 0, (-1)^n*matdet(matrix(n, n, i, j, if(abs(i-j)<2, 2+abs(i-j))))) /* Michael Somos, Sep 15 2005 */
(Magma) [(-3)^n*Evaluate(ChebyshevU(n+1), 1/3): n in [0..50]]; // G. C. Greubel, Jan 02 2024
(SageMath) [3^n*chebyshev_U(n, -1/3) for n in range(41)] # G. C. Greubel, Jan 02 2024
CROSSREFS
KEYWORD
sign,easy
AUTHOR
STATUS
approved