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%I #50 Sep 23 2023 09:04:35
%S 1,1,7,9,31,49,127,225,511,961,2047,3969,8191,16129,32767,65025,
%T 131071,261121,524287,1046529,2097151,4190209,8388607,16769025,
%U 33554431,67092481,134217727,268402689,536870911,1073676289,2147483647
%N G.f.: (1 + 2*x^2)/((1 + x)*(1 - 2*x)*(1 - 2*x^2)).
%C Resultant of the polynomial x^n - 1 and the Chebyshev polynomial of the first kind T_2(x).
%C This sequence is the case P1 = 1, P2 = 0, Q = -2 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Apr 27 2014
%H Vincenzo Librandi, <a href="/A085903/b085903.txt">Table of n, a(n) for n = 1..1000</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.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 (1,4,-2,-4).
%F a(2*n) = 2*4^n - 1, a(2*n + 1) = (2^n - 1)^2; interlaces A083420 with A060867 (squares of Mersenne numbers A000225). - _Creighton Dement_, May 19 2005
%F A107663(2*n) = a(2*n) = A083420(n). - _Creighton Dement_, May 19 2005
%F From _Peter Bala_, Apr 27 2014: (Start)
%F a(n) = (sqrt(2)^n - 1)*(sqrt(2)^n - (-1)^n)).
%F a(n) = Product_{k = 1..n} ( 2 - exp(4*k*Pi*i/n) ). (End)
%F E.g.f.: exp(-x) + exp(2*x) - 2*cosh(sqrt(2)*x). - _Ilya Gutkovskiy_, Jun 16 2016
%p seq(simplify((sqrt(2)^n - 1)*(sqrt(2)^n - (-1)^n)), n = 1..30); # _Peter Bala_, Apr 27 2014
%t CoefficientList[ Series[(1 + 2x^2)/(1 - x - 4x^2 + 2x^3 + 4x^4), {x, 0, 30}], x] (* _Robert G. Wilson v_, May 04 2013 *)
%t LinearRecurrence[{1,4,-2,-4},{1,1,7,9},40] (* _Harvey P. Dale_, Jul 25 2016 *)
%o (PARI) a(n) = polresultant(x^n - 1, 2*x^2 - 1) \\ _David Wasserman_, Feb 10 2005
%o (Magma) [Round((Sqrt(2)^n - 1)*(Sqrt(2)^n - (-1)^n)): n in [1..40]]; // _Vincenzo Librandi_, Apr 28 2014
%Y Cf. A083420, A028400, A062510, A088037, A107663.
%Y Cf. A060867. A100047.
%K nonn,easy
%O 1,3
%A Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 16 2003
%E More terms from _David Wasserman_, Feb 10 2005
%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 15 2007
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