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A331691
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Resultant of the Shapiro polynomials P_n(x) and Q_n(x).
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2
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = (-1)^(n-1) * 2^(2^(n+1) - n - 2) for n >= 1 [Brillhart and Carlitz theorem 2].
a(n) = (-1)^(n-1) * A016031(n+2) for n >= 1.
a(n) = - 2^(2^n-1) * a(n-1) for n >= 2 [Brillhart and Carlitz in proof of theorem 2].
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PROG
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(PARI) a(n) = if(n==0, 1, -(-2)^(2^(n+1) - n - 2));
(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);
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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