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%I #34 Sep 08 2022 08:45:37
%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 G. C. Greubel, <a href="/A143699/b143699.txt">Table of n, a(n) for n = 0..750</a>
%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 Equals sqrt(A003733(n)/5).
%F G.f.: x*(1+x)*(1-x)/(1 - 19*x + 41*x^2 - 19*x^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, (sqrt(5) - 9)/4)*U(n-1, -(sqrt(5) + 9)/4) 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-François Alcover_, Dec 12 2016 *)
%o (PARI) {a(n) = n = abs(n); polcoeff( x*(1-x^2)/(1 -19*x +41*x^2 -19*x^3 +x^4) + x*O(x^n), n)} \\ _Michael Somos_, Feb 24 2009
%o (Magma) I:=[0,1,19,319]; [n le 4 select I[n] else 19*Self(n-1) -41*Self(n-2) +19*Self(n-3) -Self(n-4): n in [1..30]]; // _G. C. Greubel_, May 31 2021
%o (Sage)
%o def A143699_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x*(1-x^2)/(1 -19*x +41*x^2 -19*x^3 +x^4) ).list()
%o A143699_list(30) # _G. C. Greubel_, May 31 2021
%Y Cf. A003733, 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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