A100051 A Chebyshev transform of 1,1,1,...

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

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Rational Function Multiplicative Coefficients

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

Formula

From Paul Barry, Dec 01 2004: (Start)

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. (End)

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

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

Mathematica

CoefficientList[Series[(1 - x^2)/(1 - x + x^2), {x, 0, 50}], x] (* G. C. Greubel, May 03 2017 *)
LinearRecurrence[{1, -1}, {1, 1, -1}, 80] (* Harvey P. Dale, Mar 25 2019 *)

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 * A281727 A122876 A131713

Adjacent sequences: A100048 A100049 A100050 * A100052 A100053 A100054

Keyword

easy,sign,mult

Author

Paul Barry, Oct 31 2004

Status

approved