OFFSET
0,5
COMMENTS
(Start) Let {U_N(x)}, N=0,1,2,..., be Chebyshev polynomials of the second kind, defined by U_0(x)=1, U_1(x)=2*x, U_N(x)=2*x*U_(N-1)(x)-U_(N-2)(x). Let n>2 and q_n=floor(n/2). Let A^<n>={A_(n,0),A_(n,1),...,A_(n,q_n-1)} be the n-th set of unit-primitive matrices (see [Jeffery]). We form a matrix containing the ordered spectra of the unit-primitives from A^<n> as follows. Define the column vectors S(A_(n,r))=Specrum(A_(n,r))=(U_r(x_0),U_r(x_1),...,U_r(x_(q_n-1)))^T, with x_r=cos((2*r+1)*Pi/n), r=0,1,...,q_n-1, where T denotes the transpose. Let S^<n> be the q_n X q_n matrix formed by combining the S(A_(n,r)) such that
S^<n>=[S(A_(n,0)),S(A_(n,1)),...,S(A_(n,q_n-1))]=
(1, U_1(x_0),...,U_(q_n-1)(x_0));
(1, U_1(x_1),...,U_(q_n-1)(x_1));
... (etc.)
(1, U_1(x_(q_n-1)),...,U_(q_n-1)(x_(q_n-1))),
so column r contains the (ordered) spectrum of A_(n,r). Then for our sequence, let a(0)=a(1)=0 and a(2)=a(3)=1, and, for n>3, we have
a(n)=Trace((S^<n>)^T*S^<n>). (End)
LINKS
L. E. Jeffery, Unit-primitive matrices
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).
FORMULA
G.f.: x^2*(x^3+2*x^2+x+1)/((x-1)*(x+1))^4.
a(n) = 4*a(n-2)-6*a(n-4)+4*a(n-6)-a(n-8), for n>7.
a(2*n) = A002411(n).
a(2*n+1) = A000330(n).
a(n) = n*(n*(5*n+6)+(n^2+6*n+2)*(-1)^n-2)/96. - Bruno Berselli, Apr 14 2011
EXAMPLE
Suppose n=7, then S^<7>=[(1,1.8019..,2.2469..);(1,0.445..,-0.8019..);(1,-1.2469..,0.5549..)] and (S^<7>)^T*S^<7>=[(3,1,2);(1,5,3);(2,3,6)]. Hence a(7)=Trace((S^<7>)^T*S^<7>)=3+5+6=14. (Note that (S^<n>)^T*S^<n> is always integral.)
PROG
(Maxima) makelist(coeff(taylor(x^2*(x^3+2*x^2+x+1)/((x-1)*(x+1))^4, x, 0, n), x, n), n, 0, 50); /* Bruno Berselli, May 30 2011 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
L. Edson Jeffery, Apr 11 2011
STATUS
approved