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%I #24 Sep 08 2022 08:45:49
%S 0,1,1,8,17,81,224,881,2737,9928,32481,113761,380800,1313441,4441121,
%T 15215688,51677297,176530481,600723424,2049428881,6980069457,
%U 23799693448,81088954561,276417142721,941948403200,3210574806081
%N G.f. -x*(x-1)*(1+x)/(1-x-8*x^2-x^3+x^4).
%C The member k=8 of a family of sequences starting 0,1,1,k with recurrence a(n) = a(n-1)+k*a(n-2)+a(n-3)-a(n-4).
%C This is the case P1 = 1, P2 = -10, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Mar 31 2014
%H Vincenzo Librandi, <a href="/A171065/b171065.txt">Table of n, a(n) for n = 0..1000</a>
%H Hugh Williams, R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory vol. 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
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (1,8,1,-1).
%F a(n)= +a(n-1) +8*a(n-2) +a(n-3) -a(n-4).
%F From _Peter Bala_, Mar 31 2014: (Start)
%F a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(41))/4 and beta = (1 - sqrt(41))/4 and T(n,x) denotes the Chebyshev polynomial of the first 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, 5/2; 1, 1/2].
%F a(n) = U(n-1,i*(1 + sqrt(2))/2)*U(n-1,i*(1 + sqrt(2))/2), where U(n,x) denotes the Chebyshev polynomial of the second kind.
%F See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)
%t CoefficientList[Series[-x*(x - 1)*(1 + x)/(1 - x - 8*x^2 - x^3 + x^4), {x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 19 2012 *)
%t LinearRecurrence[{1,8,1,-1},{0,1,1,8},30] (* _Harvey P. Dale_, Dec 27 2017 *)
%o (Magma) I:=[0, 1, 1, 8]; [n le 4 select I[n] else Self(n-1) + 8*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Dec 19 2012
%Y Cf. A116201 (k=1), A105309 (k=2), A152090 (k=3), A007598 (k=4), A005178 (k=5), A003757 (k=6). A100047.
%K nonn,easy
%O 0,4
%A _R. J. Mathar_, at the request of _R. K. Guy_, Sep 03 2010
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