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%I #20 Feb 06 2021 21:51:02
%S 1,70,5299,419020,33664741,2719393810,220069738519,17820217484440,
%T 1443290970139081,116902609136432350,9469004435040169339,
%U 766986472802959676260,62125826363286791503021,5032189831214900660779690,407607319514701058318401759,33016191346720726553176114480
%N A fourth-order linear divisibility sequence: a(n) := (3^n + 1)*(3^(3*n) - 1)/( (3 + 1)*(3^3 - 1)).
%C This is a divisibility sequence, that is, if n | m then a(n) | a(m). More generally, the polynomials P(n,x) := (x^n + 1)*(x^(3*n) - 1) form a sequence of divisibility polynomials in the polynomial ring Z[x]; that is, if n divides m then P(n,x) divides P(m,x) in Z[x]. See the Bala link for a proof and generalization. Here we consider the integer sequence coming from the normalized polynomials P(n,x)/P(n,1) at x = 3.
%C 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 discovered 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 parameter.
%C For sequences of a similar type see A238536 through A238541.
%H Peter Bala, <a href="/A238536/a238536.pdf">A family of linear divisibility sequences of order four</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Divisibility_sequence">Divisibility sequence</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="https://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">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 (112,-2622,9072,-6561).
%F a(n) = (1/104)*(3^n + 1)*(3^(3*n) - 1) = (1/104)*(9^n - 1)*(27^n - 1)/(3^n - 1).
%F O.g.f.: x*(1 - 42*x + 81*x^2)/((1 - x)*(1 - 3*x)*(1 - 27*x)*(1 - 81*x)).
%F Recurrence equation: a(n) = 112*a(n-1) - 2622*a(n-2) + 9072*a(n-3) - 6561*a(n-4).
%p #A238540
%p seq(1/104*(3^n + 1)*(3^(3*n) - 1), n = 1..20);
%t LinearRecurrence[{112, -2622, 9072, -6561}, {1, 70, 5299, 419020}, 16] (* _Jean-François Alcover_, Nov 14 2019 *)
%Y Cf. A238536, A238537, A238538, A238539, A238541.
%K nonn,easy
%O 1,2
%A _Peter Bala_, Mar 01 2014
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