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A238541 A fourth-order linear divisibility sequence: a(n) := A(n)/A(1) where A(n) := ( (3^n + 2^n)*(3^(3*n) - 2^(3*n)) ).

%I A238541 #13 Feb 06 2021 21:51:12
%S A238541 1,91,7063,538447,41441455,3231753343,254851186927,20265345051679,
%T A238541 1621012954550479,130194036583465855,10485834936321976111,
%U A238541 846117830539227426271,68360837263665964839823,5527792975131721247371327,447241733557623755497669615
%N A238541 A fourth-order linear divisibility sequence: a(n) := A(n)/A(1) where A(n) := ( (3^n + 2^n)*(3^(3*n) - 2^(3*n)) ).
%C A238541 The bivariate polynomials P(n,x,y) := (x^n + y^n)*(x^(3*n) - y^(3*n)) form a sequence of divisibility polynomials in the polynomial ring Z[x,y]; that is, if n divides m then P(n,x,y) divides P(m,x,y) in Z[x,y] (see the Bala link). Here we consider the integer sequence coming from the normalized polynomials P(n,x,y)/P(1,x,y) when x = 3 and y = 2. Other cases include A238538(x = 2, y = 1), A238539(x = 2, y = -1) and A238540(x = 3, y = 1). See also A238536, A238537 and A215466.
%H A238541 Peter Bala, <a href="/A238536/a238536.pdf">A family of linear divisibility sequences of order four</a>
%H A238541 Wikipedia, <a href="http://en.wikipedia.org/wiki/Divisibility_sequence">Divisibility sequence</a>
%H A238541 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (175,-10158,226800,-1679616).
%F A238541 a(n) = (1/95)*(3^n + 2^n)*(3^(3*n) - 2^(3*n)).
%F A238541 a(n) = (1/95)*(9^n - 4^n)*(27^n - 8^n)/(3^n - 2^n).
%F A238541 O.g.f.: x*(1 - 84*x + 1296*x^2)/((1 - 16*x)*(1 - 24*x)*(1 - 54*x)*(1 - 81*x)).
%F A238541 Recurrence equation: a(n) = 175*a(n-1) - 10158*a(n-2) + 226800*a(n-4) - 1679616*a(n-4).
%p A238541 #A238541
%p A238541 seq(1/95*(3^n + 2^n)*(3^(3*n) - 2^(2*n)), n = 1..20);
%t A238541 LinearRecurrence[{175,-10158,226800,-1679616},{1,91,7063,538447},20] (* _Harvey P. Dale_, Apr 12 2018 *)
%Y A238541 Cf. A215466, A238536, A238537, A238538, A238539, A238540.
%K A238541 nonn,easy
%O A238541 1,2
%A A238541 _Peter Bala_, Mar 01 2014