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A057077
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Periodic sequence 1,1,-1,-1; expansion of (1+x)/(1+x^2).
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65
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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
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OFFSET
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0,1
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COMMENTS
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Sum_{k>=0} a(k)/(k+1) = Sum_{k>=0} 1/((a(k)*(k+1))) = log(2)/2 + Pi/4 [From Jaume Oliver Lafont, Apr 30 2010]
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LINKS
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Table of n, a(n) for n=0..71.
Index to sequences with linear recurrences with constant coefficients, signature (0,-1).
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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G.f.: (1+x)/(1+x^2).
a(n) = S(n, 0)+S(n-1, 0) = S(2*n, sqrt(2)); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 0)=A056594.
a(n)=cos(n*Pi/2)+sin(n*Pi/2) with n>=0 - Paolo P. Lava, Jun 12 2006
a(n) = (-1)^binomial(n,2) = (-1)^floor(n/2) = 1/2*((n+2) mod 4 - n mod 4). For fixed r = 0,1,2,..., it appears that (-1)^binomial(n,2^r) gives a periodic sequence of period 2^(r+1), the period consisting of a block of 2^r plus ones followed by a block of 2^r minus ones. See A033999 (r = 0), A143621 (r = 2) and A143622 (r = 3). Define E(k) = sum {n = 0..inf} a(n)*n^k/n! for k = 0,1,2,... . Then E(0) = cos(1) + sin(1), E(1) = cos(1) - sin(1) and E(k) is an integral linear combination of E(0) and E(1) (a Dobinski-type relation). Precisely, E(k) = A121867(k) * E(0) - A121868(k) * E(1). See A143623 and A143624 for the decimal expansions of E(0) and E(1) respectively. For a fixed value of r, similar relations hold between the values of the sums E_r(k) := sum {n = 0..inf} (-1)^floor(n/r)*n^k/n!, k = 0,1,2,... . For particular cases see A000587 (r = 1) and A143628 (r = 3). [From Peter Bala, Aug 28 2008]
a(n)=(1/2)*[(1-I)*I^n+(1+I)*(-I)^n], with I=sqrt(-1) [From Paolo P. Lava, May 26 2010]
a(n)=(-1)^A180969(1,n), where the first index in A180969(.,.) is the row index [Adriano Caroli, Nov 18 2010]
a(n)=(-1)^((2*n+(-1)^n-1)/4) = i^(n*(n-1)), with i=sqrt(-1). - Bruno Berselli, Dec 27 2010 - Aug 26 2011
Non-simple continued fraction expansion of (3+sqrt(5))/2 = A104457. - R. J. Mathar, Mar 08 2012
E.g.f.: cos(x)*(1 + tan(x)). - Arkadiusz Wesolowski, Aug 31 2012
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PROG
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(Maxima) A057077(n) := block(
[1, 1, -1, -1][1+mod(n, 4)]
)$ /* R. J. Mathar, Mar 19 2012 */
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CROSSREFS
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|a(n)|=A000012. Cf. A049310.
Cf. A000587, A121867, A121868, A130151, A143621, A143622, A143623, A143624, A143628. [From Peter Bala, Aug 28 2008]
Sequence in context: A131554 A153881 A160357 * A070748 A154990 A209615
Adjacent sequences: A057074 A057075 A057076 * A057078 A057079 A057080
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KEYWORD
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sign,easy
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AUTHOR
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Wolfdieter Lang, Aug 04 2000
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STATUS
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approved
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