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A238539 A fourth-order linear divisibility sequence: a(n) := (1/9)*(2^n + (-1)^n)*(2^(3*n) - (-1)^n).

%I A238539 #18 Feb 06 2021 21:50:54
%S A238539 1,35,399,7735,112871,1893255,29593159,479082695,7620584391,
%T A238539 122287263175,1953732901319,31282632909255,500338874618311,
%U A238539 8006888009380295,128098480026087879,2049669505409577415,32793961486615474631,524709388585350492615,8395302178969583120839
%N A238539 A fourth-order linear divisibility sequence: a(n) := (1/9)*(2^n + (-1)^n)*(2^(3*n) - (-1)^n).
%C A238539 This is a divisibility sequence, that is, if n | m then a(n) | a(m). This is a consequence of the following more general result: The polynomials P(n,x,y) := (x^n + y^n)*(x^(3*n) - y^(3*n)) form a divisibility sequence in the polynomial ring Z[x,y]. See the Bala link.
%C A238539 The sequence satisfies a homogeneous linear recurrence of the fourth order. However, it does not belong to the family of linear divisibility sequences of the fourth order studied by Williams and Guy, which have o.g.f.s of the form x*(1 - q*x^2)/Q(x), Q(x) a quartic polynomial and q an integer.
%C A238539 For sequences of a similar type see A238536 through A238541.
%H A238539 Peter Bala, <a href="/A238536/a238536.pdf">A family of linear divisibility sequences of order four</a>
%H A238539 Wikipedia, <a href="http://en.wikipedia.org/wiki/Divisibility_sequence">Divisibility sequence"</a>
%H A238539 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 A238539 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 A238539 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,138,112,-256).
%F A238539 a(n) = (1/9)*(2^n + (-1)^n)*(2^(3*n) - (-1)^n) = (1/9)*(4^n - 1)*(8^n - (-1)^n)/(2^n - (-1)^n).
%F A238539 O.g.f.: x*(1 + 28*x + 16*x^2)/((1 - x)*(1 + 2*x)*(1 + 8*x)*(1 - 16*x)).
%F A238539 Recurrence equation: a(n) = 7*a(n-1) + 138*a(n-2) + 112*a(n-4) - 256*a(n-4).
%p A238539 seq(1/9*(2^n + (-1)^n)*(2^(3*n) - (-1)^n), n = 1..20);
%Y A238539 Cf. A238536, A238537, A238538, A238540, A238541.
%K A238539 nonn,easy
%O A238539 1,2
%A A238539 _Peter Bala_, Mar 01 2014