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A112712 Expansion of x/(1 - x + 2*x^2 - 2*x^3 + 2*x^4 - x^5 + x^6). 3
0, 1, 1, -1, -1, 1, 0, -2, 0, 2, 0, -1, 1, 1, -1, -1, 0, 0, 0, 0, 0, 1, 1, -1, -1, 1, 0, -2, 0, 2, 0, -1, 1, 1, -1, -1, 0, 0, 0, 0, 0, 1, 1, -1, -1, 1, 0, -2, 0, 2, 0, -1, 1, 1, -1, -1, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

A modified Chebyshev transform of the Fibonacci numbers F(n) under the mapping g(x) -> (1/(1 + x^2)^2)*g(x/(1 + x^2)).

LINKS

Table of n, a(n) for n=0..59.

The Knot Atlas, L6a2

Index entries for linear recurrences with constant coefficients, signature (1,-2,2,-2,1,-1).

FORMULA

G.f.: x*(1 + x - x^2 - x^3 + x^4 - 2*x^6 + 2*x^8 - x^10 + x^11 + x^12 - x^13 - x^14)/(1 - x^20).

a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + a(n-5) - a(n-6).

a(n) = a(n-20).

a(n) = Sum_{k=0..floor((n+2)/2)} (-1)^(k + 1)*C(n - k + 2, k - 1)*F(n-2k+2).

a(n) = Sum_{k=0..n} (F(k)*(-1)^((n - k)/2)*(Sum{j=0..n} C((j + k)/2, k)*(1 + (-1)^(n - j))(1 + (-1)^(j - k))/4)).

G.f.: -1/(x^(13/2)*f(x)), where f(x) = -1/x^(3/2) + 1/x^(5/2) - 2/x^(7/2) + 2/x^(9/2) - 2/x^(11/2) + 1/x^(13/2) - 1/x^(15/2) is the Jones polynomial for the link with Dowker-Thistlethwaite notation L6a2. - Roger L. Bagula, Jun 06 2007

MATHEMATICA

f[x_] = -1/x^(3/2) + 1/x^(5/2) - 2/x^(7/2) + 2/x^(9/2) - 2/x^(11/2) + 1/x^(13/2) - 1/x^(15/2);

CoefficientList[Series[-(1/(x^(13/2)*f[x])), {x, 0, 50}], x] (* Roger L. Bagula, Jun 06 2007 *)

LinearRecurrence[{1, -2, 2, -2, 1, -1}, {0, 1, 1, -1, -1, 1}, 80] (* Harvey P. Dale, Jun 14 2019 *)

PROG

(Maxima) a(n) := sum((-1)^(k + 1)*binomial(n - k + 2, k - 1)*fib(n - 2*k + 2), k, 0, floor((n + 2)/2)); /* Franck Maminirina Ramaharo, Jan 08 2019 */

CROSSREFS

Cf. A112713.

Sequence in context: A277142 A240592 A001877 * A026608 A264049 A287337

Adjacent sequences:  A112709 A112710 A112711 * A112713 A112714 A112715

KEYWORD

easy,sign

AUTHOR

Paul Barry, Sep 15 2005

STATUS

approved

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Last modified August 25 05:19 EDT 2019. Contains 326318 sequences. (Running on oeis4.)