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A230074
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Period 4: repeat [1, -2, 1, 0].
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1
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1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2
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OFFSET
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1,2
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COMMENTS
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The o.g.f. for this sequence is obtained from the o.g.f.'s of the bisection of the sequence including a(0) = 0.
For the cos product formula below use Product_{k=1..n-1} 2*cos(2*k*Pi/n) = 1 if n is odd, and Product_{k=1..n-1} 2*cos(2*k*Pi/n) = -(1-(-1)^(n/2)) if n is even (see Gradstein-Rhyzik, p.62, 1.393 1., with x=0).
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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) = 1 if n is odd, and a(n) = -(1-(-1)^(n/2)) if n is even.
a(n+4*k) = a(n), n = 1, 2, 3, 4, k >= 1.
G.f.: -2*x/(1-x^4) + 1/(1-x^2) = (1-x)/((1+x)*(1+x^2)).
a(n) = Product_{k=1..n-1} 2*cos(2*k*Pi/n).
a(n) + a(n-1) + a(n-2) + a(n-3) = 0, n>3.
a(n) = (1+(-1)^n)*(-1)^(n/2)/2-(-1)^n. (End)
a(n) = a(n-4) for n>4.
a(n) = cos(n*Pi/2) - (-1)^n. (End)
Multiplicative with a(2^e) = (-2)^e if e<2 and 0 if e>1, and a(p^e) = 1 for prime p > 2.
Dirichlet g.f.: zeta(s) * (1-2^(-s)) * (1-2^(1-s)).
Dirichlet inverse b(n) is multiplicative with b(2^e) = 2^e and, for prime p>2, b(p^e) = (-1)^e if e<2 and 0 if e>1. (End)
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MAPLE
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MATHEMATICA
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Table[Sqrt[Mod[n^2, 8]](-1)^(n+1), {n, 100}] (* Wesley Ivan Hurt, Jan 01 2014 *)
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PROG
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CROSSREFS
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KEYWORD
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sign,easy,mult
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AUTHOR
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STATUS
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approved
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