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A331691 Resultant of the Shapiro polynomials P_n(x) and Q_n(x).

%I A331691 #8 Feb 22 2020 19:27:58
%S A331691 1,2,-16,2048,-67108864,144115188075855872,
%T A331691 -1329227995784915872903807060280344576,
%U A331691 226156424291633194186662080095093570025917938800079226639565593765455331328
%N A331691 Resultant of the Shapiro polynomials P_n(x) and Q_n(x).
%C A331691 The Shapiro polynomials P_n(x) and Q_n(x) are defined by P_0(x) = Q_0(x) = 1 and then mutual recurrences P_{n+1}(x) = P_n(x) + x^(2^n)*Q_n(x) and Q_{n+1}(x) = P_n(x) - x^(2^n)*Q_n(x).  The coefficients of P are the Golay-Rudin-Shapiro sequence A020985.  a(n) is the polynomial resultant R(P_n(x),Q_n(x)) as considered by Brillhart and Carlitz.
%H A331691 John Brillhart and L. Carlitz, <a href="https://doi.org/10.1090/S0002-9939-1970-0260955-6">Note on the Shapiro Polynomials</a>, Proceedings of the American Mathematical Society, volume 25, number 1, May 1970, pages 114-118.  Also <a href="http://www.jstor.org/stable/2036537">at JSTOR</a>.  See A001782 for a scanned copy.
%H A331691 Harold S. Shapiro, <a href="http://hdl.handle.net/1721.1/12198">Extremal Problems for Polynomials and Power Series</a>, Masters Thesis, Massachusetts Institute of Technology, 1951.  See pages 40-41.
%F A331691 a(n) = (-1)^(n-1) * 2^(2^(n+1) - n - 2) for n >= 1 [Brillhart and Carlitz theorem 2].
%F A331691 a(n) = (-1)^(n-1) * A016031(n+2) for n >= 1.
%F A331691 a(n) = - 2^(2^n-1) * a(n-1) for n >= 2 [Brillhart and Carlitz in proof of theorem 2].
%o A331691 (PARI) a(n) = if(n==0,1, -(-2)^(2^(n+1) - n - 2));
%o A331691 (PARI) a(n) = my(P=1,Q=1); for(i=0,n-1, [P,Q]=[P+x^(2^i)*Q, P-x^(2^i)*Q]); polresultant(P,Q);
%Y A331691 Cf. A016031 (absolute values), A001782 (discriminant).
%K A331691 sign,easy
%O A331691 0,2
%A A331691 _Kevin Ryde_, Jan 24 2020