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%I #29 Dec 07 2019 12:18:24
%S -1,-3,0,21,55,0,-377,-987,0,6765,17711,0,-121393,-317811,0,2178309,
%T 5702887,0,-39088169,-102334155,0,701408733,1836311903,0,-12586269025,
%U -32951280099,0,225851433717,591286729879,0,-4052739537881
%N Expansion of (x-1)*(x+1) / (8*x^2 + 1 - 3*x + x^4 - 3*x^3).
%C Terms (or their respective absolute values) appear to be contained in A000045.
%C Working with an offset of 1, this sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = 3, P2 = 6, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Mar 25 2014
%H Vincenzo Librandi, <a href="/A108196/b108196.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-8,3,-1).
%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="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.pdf">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume
%F a(0)=-1, a(1)=-3, a(2)=0, a(3)=21, a(n) = 3*a(n-1) - 8*a(n-2) + 3*a(n-3) - a(n-4). - _Harvey P. Dale_, Dec 25 2012
%F From _Peter Bala_, Mar 25 2014: (Start)
%F The following formulas assume an offset of 1.
%F a(n) = (-1)*A001906(n)*A010892(n-1). Equivalently, a(n) = (-1)*U(n-1,1/2)*U(n-1,3/2), where U(n,x) denotes the Chebyshev polynomial of the second kind.
%F a(n) = (-1)*bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -3/2; 1, 3/2] and T(n,x) denotes the Chebyshev polynomial of the first kind.
%F The ordinary generating function is the Hadamard product of -x/(1 - x + x^2) and x/(1 - 3*x + x^2).
%F See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
%p seriestolist(series((x-1)*(x+1)/(8*x^2+1-3*x+x^4-3*x^3), x=0,40));
%t CoefficientList[Series[(x-1)(x+1)/(8x^2+1-3x+x^4-3x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,-8,3,-1},{-1,-3,0,21},40] (* _Harvey P. Dale_, Dec 25 2012 *)
%o (Sage) [lucas_number1(n,3,1)*lucas_number1(n,1,1)*(-1) for n in range(1,33)] # _Zerinvary Lajos_, Jul 06 2008
%o (PARI) x='x+O('x^50); Vec((x-1)*(x+1)/(8*x^2 +1 -3*x + x^4 - 3*x^3)) \\ _G. C. Greubel_, Aug 08 2017
%Y Cf. A000045, A100047.
%K easy,sign
%O 0,2
%A _Creighton Dement_, Jul 23 2005
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