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A161158 A003696(n+1)/A001353(n+1). 3

%I

%S 1,14,161,1792,19809,218638,2412353,26614784,293628097,3239445006,

%T 35739069409,394290020096,4349990523425,47991114171406,

%U 529460241815169,5841251080892416,64443392518654337,710969410782059534

%N A003696(n+1)/A001353(n+1).

%C Proposed by R. Guy in the seqfan list Mar 28 2009.

%C With an offset of 1, this sequence is the case P1 = 14, P2 = 32, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Apr 27 2014

%H Vincenzo Librandi, <a href="/A161158/b161158.txt">Table of n, a(n) for n = 0..900</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

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (14, -34, 14, -1).

%F a(n)= 14*a(n-1) -34*a(n-2) +14*a(n-3) -a(n-4). G.f.: (1-x^2)/(1-14*x+34*x^2-14*x^3+x^4).

%F From _Peter Bala_, Apr 27 2014: (Start)

%F The following remarks assume an offset of 1.

%F a(n) = (1/sqrt(17))*( T(n,(7 + sqrt(17))/2) - T(n,(7 - sqrt(17))/2) ), where T(n,x) is 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, -8; 1, 7].

%F a(n) = U(n-1,1/2*(4 + sqrt(2)))*U(n-1,1/2*(4 - sqrt(2))), where U(n,x) is the Chebyshev polynomial of the second 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 CoefficientList[Series[(1 - x^2)/(1 - 14 x + 34 x^2 -14 x^3 + x^4), {x, 0, 20}], x] (* _Vincenzo Librandi_, Apr 28 2014 *)

%o (MAGMA) I:=[1,14,161,1792]; [n le 4 select I[n] else 14*Self(n-1)-34*Self(n-2)+14*Self(n-3)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Apr 28 2014

%Y Cf. A001563, A003696, A100047.

%K nonn,easy

%O 0,2

%A _R. J. Mathar_, Jun 03 2009

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Last modified October 20 22:44 EDT 2019. Contains 328291 sequences. (Running on oeis4.)