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A100051 A Chebyshev transform of 1,1,1,... 12
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; text; internal format)
OFFSET

0,4

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, Dec 01 2004

Multiplicative with a(p^e) = -1 if p = 2; -2 if p = 3; 1 otherwise. - David W. Wilson Jun 10 2005

LINKS

Table of n, a(n) for n=0..74.

M. Somos, Rational Function Multiplicative Coefficients

Index entries for linear recurrences with constant coefficients, signature (1,-1).

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, 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. - R. J. Mathar, 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

EXAMPLE

1 + x - x^2 - 2*x^3 - x^4 + x^5 + 2*x^6 + x^7 - x^8 - 2*x^9 - x^10 + ...

PROG

(PARI) {a(n) = - (n == 0) + [ 2, 1, -1, -2, -1, 1][n%6 + 1]} /* Michael Somos, Mar 21 2011 */

CROSSREFS

Cf. A099837, A099443, A011655, A100047, A100048, A100050.

Row sums of array A127677.

Sequence in context: A205375 A016010 A099837 * A122876 A131713 A100063

Adjacent sequences:  A100048 A100049 A100050 * A100052 A100053 A100054

KEYWORD

easy,sign,mult

AUTHOR

Paul Barry, Oct 31 2004

STATUS

approved

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Last modified December 8 12:55 EST 2016. Contains 278945 sequences.