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A087960
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a(n) = (-1)^binomial(n+1,2).
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22
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1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1
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OFFSET
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0,1
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COMMENTS
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Period 4: repeat [1, -1, -1, 1]. - Joerg Arndt, Feb 14 2016
Also equal to the sign of product(j-i, 1<=j<i<=n+1) = the sign of the Vandermonde determinant for -1, -2, ..., -(n+1).
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REFERENCES
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I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.
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LINKS
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FORMULA
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a(n) = Product_{k=1..n} 2*cos(2*k*Pi/(2*n+1)) for n>=0 (for n=0 the empty product is put to 1). See the Gradstein-Ryshik reference, p. 63, 1.396 2. with x = sqrt(-1). - Wolfdieter Lang, Oct 22 2013
a(n) + a(n-2) = 0 for n>1, a(n) = a(n-4) for n>3. - Wesley Ivan Hurt, Jul 07 2016
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EXAMPLE
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a(1) = -1 since (-1)^binomial(2,2) = (-1)^1 = -1.
G.f. = 1 - x - x^2 + x^3 + x^4 - x^5 - x^6 + x^7 + x^8 - x^9 - x^10 + ...
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MAPLE
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MATHEMATICA
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(-1)^Binomial[Range[0, 110], 2] (* or *) LinearRecurrence[{0, -1}, {1, 1}, 110] (* Harvey P. Dale, Jul 07 2014 *)
a[ n_] := (-1)^Quotient[ n + 1, 2]; (* Michael Somos, Jul 20 2015 *)
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PROG
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(PARI) {a(n) = (-1)^((n + 1)\2)}; /* Michael Somos, Jul 20 2015 */
(Haskell)
a087960 n = (-1) ^ (n * (n + 1) `div` 2)
(Magma) [(-1)^Binomial(n+1, 2) : n in [0..100]]; // Wesley Ivan Hurt, Jul 07 2016
(Python)
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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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EXTENSIONS
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Offset and Vandermonde formula corrected by R. J. Mathar, Sep 25 2009
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STATUS
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approved
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