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A143699 a(n) = 19*a(n-1) - 41*a(n-2) + 19*a(n-3) - a(n-4). 6

%I

%S 0,1,19,319,5301,88000,1460701,24245719,402446619,6680076601,

%T 110880352000,1840465787401,30549274537419,507077165538919,

%U 8416803858813901,139707705280792000,2318961358994380101

%N a(n) = 19*a(n-1) - 41*a(n-2) + 19*a(n-3) - a(n-4).

%C This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m).

%C A003733 = 5 * (A143699)^2. - _R. K. Guy_, Mar 11 2010

%C The sequence is the case P1 = 19, P2 = 39, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Apr 03 2014

%H Per Hakan Lundow, <a href="http://www.theophys.kth.se/~phl/Text/1factors2.pdf">Enumeration of matchings in polygraphs</a>, Section 8.1.

%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/Di#divseq">Index to divisibility sequences</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (19,-41,19,-1).

%F G.f.: x(1+x)(1-x)/(1-19x+41x^2-19x^3+x^4). - _R. J. Mathar_, Feb 09 2009

%F a(-n) = a(n). - _Michael Somos_, Feb 24 2009

%F a(n) = (r1^n + r2^n - r3^n - r4^n) / s1 where s1 = sqrt(205), s2 = sqrt(550 + 38*s1), s3 = 36 * sqrt(5) / s2, r1 = (19 + s1 + s2) / 4, r2 = 1/r1, r3 = (19 - s1 + s3) / 4, r4 = 1/r3. - _Michael Somos_, Feb 12 2012

%F From _Peter Bala_, Apr 03 2014: (Start)

%F a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), n >= 1, where alpha = 1/4*(19 + sqrt(205)), beta = 1/4*(19 - sqrt(205)) and where T(n,x) denotes the Chebyshev polynomial of the first kind.

%F a(n)= U(n-1,1/4*(sqrt(5) - 9))*U(n-1,1/4*(- sqrt(5) - 9)) 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, -39/4; 1, 19/2]. See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)

%t LinearRecurrence[{19, -41, 19, -1}, {0, 1, 19, 319}, 20] (* _Jean-Fran├žois Alcover_, Dec 12 2016 *)

%o (PARI) {a(n) = n = abs(n); polcoeff( (x - x^3) / (1 - 19*x + 41*x^2 - 19*x^3 + x^4) + x * O(x^n), n)} /* _Michael Somos_, Feb 24 2009 */

%Y Equals sqrt(A003733(n)/5). A100047.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, based on email from _R. K. Guy_, Feb 08 2009

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Last modified November 19 08:43 EST 2019. Contains 329318 sequences. (Running on oeis4.)