A099918 A Chebyshev transform related to the 7th cyclotomic polynomial.

1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1

Offset

0,3

Comments

The g.f. is a Chebyshev transform of 1/(1+x-2x^2-x^3) under the Chebyshev mapping g(x)->(1/(1+x^2))g(x/(1+x^2)). The denominator is the 7th cyclotomic polynomial. The inverse of the 7 cyclotomic polynomial A014016 is given by sum{k=0..n, A099918(n-k)(k/2+1)(-1)^(k/2)(1+(-1)^k)/2}.

Links

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

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

Formula

G.f.: (1+x^2)^2/(1+x+x^2+x^3+x^4+x^5+x^6); a(n)=sum{k=0..floor(n/2), C(n-k, k)^(-1)^k*b(n-2k)}, where b(n)=A094790(n/2+1)(1+(-1)^n)/2+A094789((n+1)/2)(1-(-1)^n)/2=(-1)^n*A006053(n+2).

a(n)=(1/7)*{-(n mod 7)-[(n+1) mod 7]+2*[(n+2) mod 7]-3*[(n+3) mod 7]+4*[(n+4) mod 7]-3*[(n+5) mod 7]+2*[(n+6) mod 7]}, Paolo P. Lava, Mar 10 2011

Mathematica

LinearRecurrence[{-1, -1, -1, -1, -1, -1}, {1, -1, 2, -2, 1, -1}, 90] (* Harvey P. Dale, May 25 2019 *)

Crossrefs

Cf. A099860.

Sequence in context: A317529 A285194 A039978 * A099860 A317950 A255212

Adjacent sequences: A099915 A099916 A099917 * A099919 A099920 A099921

Keyword

easy,sign

Author

Paul Barry, Oct 30 2004

Status

approved