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 A152090 a(n) = 2^n*Product_{k=1..floor((n-1)/2)} (1 + 2*cos(k*Pi/n)^2 + 4*cos(k*Pi/n)^4). 13

%I

%S 1,1,1,3,7,16,39,91,217,513,1216,2881,6825,16171,38311,90768,215047,

%T 509491,1207089,2859841,6775552,16052673,38032081,90105811,213479175,

%U 505776016,1198287271,2838988683,6726147337,15935624641,37754768064

%N a(n) = 2^n*Product_{k=1..floor((n-1)/2)} (1 + 2*cos(k*Pi/n)^2 + 4*cos(k*Pi/n)^4).

%C Limiting ratio after n=30 terms is 2.369205407038926.

%C With a(0)=0, this is a divisibility sequence with g.f. x(1-x^2)/(1 - x - 3x^2 - x^3 + x^4). The limiting ratio is the largest zero of 1 - x - 3x^2 - x^3 + x^4. - _T. D. Noe_, Dec 22 2008

%C The sequence is the case P1 = 1, P2 = -5, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Mar 25 2014

%H Peter Bala, <a href="/A100047/a100047.pdf">Linear divisibility sequences and Chebyshev polynomials</a>

%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

%H H. C. Williams and R. K. Guy, <a href="https://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,1,-1).

%F From _Colin Barker_, Jan 05 2014: (Start)

%F a(n) = a(n-1) +3*a(n-2) +a(n-3) -a(n-4) for n>4.

%F G.f.: (x^4-2*x^3-3*x^2+1) / (x^4-x^3-3*x^2-x+1). (End)

%F From _Peter Bala_, Mar 25 2014: (Start)

%F a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), n >= 1, where alpha = 1/4*(1 + sqrt(21)), beta = 1/4*(1 - sqrt(21)) and where T(n,x) denotes the Chebyshev polynomial of the first kind.

%F a(n) = U(n-1,1/4*(1 + sqrt(-3)))*U(n-1,1/4*(1 - sqrt(-3))) for n >= 1, where U(n,x) denotes the Chebyshev polynomial of the second kind.

%F a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 5/4; 1, 1/2]. See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)

%t bb = Table[FullSimplify[ExpandAll[Product[1 + 4*Cos[k*Pi/n]^2 + 16*Cos[k*Pi/n]^4, {k, 1, (n - 1)/2}]]], {n, 0, 30}]

%t LinearRecurrence[{1, 3, 1, -1}, {1, 1, 1, 3, 7}, 50] (* _G. C. Greubel_, Aug 08 2017 *)

%o (PARI) Vec((x^4-2*x^3-3*x^2+1)/(x^4-x^3-3*x^2-x+1) + O(x^100)) \\ _Colin Barker_, Jan 05 2014

%Y Cf. A100047.

%K nonn,easy

%O 0,4

%A _Roger L. Bagula_ and _Gary W. Adamson_, Nov 23 2008

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Last modified November 13 23:01 EST 2019. Contains 329106 sequences. (Running on oeis4.)