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A238540 A fourth-order linear divisibility sequence: a(n) := (3^n + 1)*(3^(3*n) - 1)/( (3 + 1)*(3^3 - 1)).

%I A238540 #20 Feb 06 2021 21:51:02
%S A238540 1,70,5299,419020,33664741,2719393810,220069738519,17820217484440,
%T A238540 1443290970139081,116902609136432350,9469004435040169339,
%U A238540 766986472802959676260,62125826363286791503021,5032189831214900660779690,407607319514701058318401759,33016191346720726553176114480
%N A238540 A fourth-order linear divisibility sequence: a(n) := (3^n + 1)*(3^(3*n) - 1)/( (3 + 1)*(3^3 - 1)).
%C A238540 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 A238540 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 A238540 For sequences of a similar type see A238536 through A238541.
%H A238540 Peter Bala, <a href="/A238536/a238536.pdf">A family of linear divisibility sequences of order four</a>
%H A238540 Wikipedia, <a href="http://en.wikipedia.org/wiki/Divisibility_sequence">Divisibility sequence</a>
%H A238540 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 A238540 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 A238540 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (112,-2622,9072,-6561).
%F A238540 a(n) = (1/104)*(3^n + 1)*(3^(3*n) - 1) = (1/104)*(9^n - 1)*(27^n - 1)/(3^n - 1).
%F A238540 O.g.f.: x*(1 - 42*x + 81*x^2)/((1 - x)*(1 - 3*x)*(1 - 27*x)*(1 - 81*x)).
%F A238540 Recurrence equation: a(n) = 112*a(n-1) - 2622*a(n-2) + 9072*a(n-3) - 6561*a(n-4).
%p A238540 #A238540
%p A238540 seq(1/104*(3^n + 1)*(3^(3*n) - 1), n = 1..20);
%t A238540 LinearRecurrence[{112, -2622, 9072, -6561}, {1, 70, 5299, 419020}, 16] (* _Jean-François Alcover_, Nov 14 2019 *)
%Y A238540 Cf. A238536, A238537, A238538, A238539, A238541.
%K A238540 nonn,easy
%O A238540 1,2
%A A238540 _Peter Bala_, Mar 01 2014