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A100051
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A Chebyshev transform of 1,1,1,...
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10
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1, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| A Chebyshev transform of 1/(1-x): if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).
Transform of 1/(1+x) under the mapping g(x)->((1+x)/(1-x))g(x/(1-x)^2). - Paul Barry (pbarry(AT)wit.ie), Dec 01 2004
Multiplicative with a(p^e) = -1 if p = 2; -2 if p = 3; 1 otherwise. David W. Wilson (davidwwilson(AT)comcast.net) Jun 10, 2005.
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LINKS
| M. Somos, Rational Function Multiplicative Coefficients
Index to sequences with linear recurrences with constant coefficients, signature (1,-1).
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FORMULA
| G.f.: (1-x^2)/(1-x+x^2); a(n)=a(n-1)-a(n-2), n>2; a(n)=n*sum{k=0..floor(n/2), (-1)^k*binomial(n-k, k)/(n-k)}.
a(n)=sum{k=0..n, binomial(n+k, 2k)(2n/(n+k))(-1)^k}, n>1 - Paul Barry (pbarry(AT)wit.ie), Dec 01 2004
Moebius transform is length 6 sequence [ 1, -2, -3, 0, 0, 6].
Euler transform of length 6 sequence [ 1, -2, -1, 0, 0, 1].
a(n) = a(-n). a(n) = c_6(n) if n>1 where c_k(n) is Ramanujan's sum. - Michael Somos Mar 21 2011
a(n)=A087204(n), n>0. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 02 2008]
a(n) = A057079(n+1), n>0. Dirichlet g.f. zeta(s) *(1-2^(1-s)-3^(1-s)+6^(1-s)). - R. J. Mathar, Apr 11 2011
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EXAMPLE
| 1 + x - x^2 - 2*x^3 - x^4 + x^5 + 2*x^6 + x^7 - x^8 - 2*x^9 - x^10 + ...
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PROG
| (PARI) {a(n) = - (n == 0) + [ 2, 1, -1, -2, -1, 1][n%6 + 1]} /* Michael Somos Mar 21 2011 */
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CROSSREFS
| Cf. A099837, A099443, A011655, A100047, A100048, A100050.
Row sums of array A127677.
Sequence in context: A016010 A131713 A099837 * A122876 A100063 A057079
Adjacent sequences: A100048 A100049 A100050 * A100052 A100053 A100054
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KEYWORD
| easy,sign,mult
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Oct 31 2004
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