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A098306 Unsigned member r=-6 of the family of Chebyshev sequences S_r(n) defined in A092184. 4

%I

%S 0,1,6,49,384,3025,23814,187489,1476096,11621281,91494150,720331921,

%T 5671161216,44648957809,351520501254,2767515052225,21788599916544,

%U 171541284280129,1350541674324486,10632792110315761,83711795208201600

%N Unsigned member r=-6 of the family of Chebyshev sequences S_r(n) defined in A092184.

%C ((-1)^(n+1))*a(n) = S_{-6}(n), n>=0, defined in A092184.

%C This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = 6, P2 = -16, Q = 1 of the 3 parameter family of 4th order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Mar 25 2014

%H G. C. Greubel, <a href="/A098306/b098306.txt">Table of n, a(n) for n = 0..1000</a>

%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/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 (7,7,-1).

%F a(n) = (T(n, 4)-(-1)^n)/5, with Chebyshev's polynomials of the first kind evaluated at x=4: T(n, 4)=A001091(n)=((4+sqrt(15))^n + (4-sqrt(15))^n)/2.

%F a(n) = 8*a(n-1) - a(n-2) + 2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.

%F a(n) = 7*a(n-1) + 7*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=6.

%F G.f.: x*(1-x)/((1+x)*(1-8*x+x^2)) = x*(1-x)/(1-7*x-7*x^2+x^3) (from the Stephan link, see A092184).

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

%F a(2*n) = 6*A001090(n)^2; a(2*n+1) = A070997(n)^2.

%F a(n) = |u(n)|^2, where {u(n)} is the Lucas sequence in the quadratic integer ring Z[sqrt(-6)] defined by the recurrence u(0) = 0, u(1) = 1, u(n) = sqrt(-6)*u(n-1) - u(n-2) for n >= 2.

%F Equivalently, a(n) = U(n-1,sqrt(-6)/2)*U(n-1,-sqrt(-6)/2), where U(n,x) denotes the Chebyshev polynomial of the second kind.

%F a(n) = 1/10*( (4 + sqrt(15))^n + (4 - sqrt(15))^n - 2*(-1)^n ).

%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, 4; 1, 3] and T(n,x) denotes the Chebyshev polynomial of the first kind.

%F See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)

%t a[n_] := 1/10*((4 + Sqrt[15])^n + (4 - Sqrt[15])^n - 2*(-1)^n) // Simplify; Table[a[n], {n, 0, 20}] (* _Jean-Fran├žois Alcover_, Apr 28 2017 *)

%t LinearRecurrence[{7, 7, -1}, {0, 1, 6, 49, 384, 3025}, 50] (* _G. C. Greubel_, Aug 08 2017 *)

%o (PARI) x='x+O('x^50); Vec(x*(1-x)/((1+x)*(1-8*x+x^2))) \\ _G. C. Greubel_, Aug 08 2017

%Y Cf. A001090, A001091, A070997, A092184, A100047.

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, Oct 18 2004

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Last modified June 20 10:13 EDT 2019. Contains 324234 sequences. (Running on oeis4.)