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%I #21 Nov 09 2021 18:02:25
%S 1,15,219,3315,51491,811395,12882499,205321155,3278747331,52408827075,
%T 838132189379,13406842675395,214483303960771,3431523432591555,
%U 54902699475185859,878429788032676035,14054769379960303811,224875452250864496835,3598000373385828511939
%N A fourth-order linear divisibility sequence: a(n) = (2^n + 1)*(2^(3*n) - 1)/ ( (2 + 1)*(2^3 - 1) ).
%C This is a fourth-order linear divisibility sequence, that is, the sequence satisfies a linear recurrence of order 4 and 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 fourth-order linear divisibility sequence in the polynomial ring Z[x,y]. See the Bala link.
%C Hence, for a fixed integers M and N, the normalized sequence (M^n + N^n)*(M^(3*n) - N^(3*n))/ ( (M + N)*(M^3 - N^3) ) for n = 1,2,3,... is a linear divisibility sequence of order 4. It has the rational o.g.f. x*(1 - 2*M*N*(M^2 - M*N + N^2)*x + (M*N)^4*x^2)/( (1 - M^4*x)*(1 - M^3*N*x)*(1 - M*N^3*x)*(1 - N^4*x) ). This is the case M = 2, N = 1. For other cases see A238539(M = 2, N = -1), A238540(M = 3, N = 1) and A238541(M = 3, N = 2). See also A238536, A238537 and A215466.
%C Note, these sequences do 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 parameter.
%H Michael De Vlieger, <a href="/A238538/b238538.txt">Table of n, a(n) for n = 1..831</a>
%H Peter Bala, <a href="/A238536/a238536.pdf">A family of linear divisibility sequences of order four</a>
%H E. L. Roettger and H. C. Williams, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Roettger/roettger12.html">Appearance of Primes in Fourth-Order Odd Divisibility Sequences</a>, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Divisibility_sequence">Divisibility sequence</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence">Lucas 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="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/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (27,-202,432,-256).
%F a(n) = (1/21)*(2^n + 1)*(2^(3*n) - 1) = A000051(n)*A024088(n)/21.
%F a(n) = (1/21)*(4^n - 1)*(8^n - 1)/(2^n - 1).
%F O.g.f.: x*(1 - 12*x + 16*x^2)/((1 - x)*(1 - 2*x)*(1 - 8*x)*(1 - 16*x)).
%F Recurrence equation: a(n) = 27*a(n-1) - 202*a(n-2) + 432*a(n-4) - 256*a(n-4).
%p seq(1/21*(2^n + 1)*(2^(3*n) - 1), n = 1..20);
%t LinearRecurrence[{27,-202,432,-256},{1,15,219,3315},20] (* _Harvey P. Dale_, Jul 04 2019 *)
%Y Cf. A215466, A236536, A236537, A238539, A238540, A238541.
%K nonn,easy
%O 1,2
%A _Peter Bala_, Feb 28 2014
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