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A078070 Expansion of (1-x)/(1 + 2*x + 2*x^2 + x^3). 6
1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Period 6: repeat [1, -3, 4, -3, 1, 0].

The unsigned sequence is the r=3 member in the r-family of sequences S_r(n) defined in A092184 where more information can be found. |a(n)| = 2-2*T(n,1/2), with twice the Chebyshev's polynomials of the first kind 2*T(n,x=1/2) = A057079(n+1) = S(n+1,1) + S(n,1) with S(n,1)= A010892(n).

Working with an offset of 1, this sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = -3, P2 = 2, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Peter Bala, Linear divisibility sequences and Chebyshev polynomials

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.

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

Index entries for sequences related to Chebyshev polynomials.

FORMULA

abs(a(n)) = 2 + 2cos(Pi*n/3 - 2Pi/3). - Paul Barry, Mar 14 2004

Euler transform of finite sequence [-3, 1, 1]. - Michael Somos, Sep 17 2004

a(n) = (n+1)sum{k=0..floor((n+1)/2), (-1)^k*binomial(n-k+1, k)*(-1)^(n-2k+1)/(n-k+1)}+2*(-1)^n; a(n)=2T(n+1, -1/2)+2(-1)^n. - Paul Barry, Dec 12 2004

From Peter Bala, Mar 25 2014: (Start)

The following formulas assume an offset of 1.

Let {u_j(n)}, j = 0 or j = 1, be two Lucas sequences in the quadratic integer ring Z[w], where w = exp(2*Pi*i/3), defined by the recurrences u_j(0) = 0, u_j(1) = 1 and u_j(n) = (-1)^j*sqrt(3)*u(n-1) - u(n-2) for n >= 2. Then a(n) = u_0(n)*u_1(n).

Equivalently, a(n) = U(n-1,sqrt(3)/2)*U(n-1,-sqrt(3)/2), where U(n,x) denotes the Chebyshev polynomial of the second kind.

a(n) = - ( ((sqrt(3) + i)/2)^n - ((sqrt(3) - i)/2)^n )*( ((-sqrt(3) + i)/2)^n - ((-sqrt(3) - i)/2)^n ) = w^n + w^(2*n) - 2*(-1)^n = 2*cos(2*n*Pi/3) - 2*(-1)^n.

a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -1/2; 1, -3/2] and T(n,x) denotes the Chebyshev polynomial of the first kind.

See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th order linear divisibility sequences. (End)

a(n) = a(n+6) = a(-2-n) for all n in Z. - Michael Somos, Aug 05 2015

a(n) = (-1)^n * A254745(n). - Michael Somos, Jul 16 2017

EXAMPLE

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

MATHEMATICA

a[ n_] := {-3, 4, -3, 1, 0, 1}[[Mod[ n, 6, 1]]]; (* Michael Somos, Aug 05 2015 *)

CoefficientList[Series[(1-x)/(1+2x+2x^2+x^3), {x, 0, 120}], x] (* or *) PadRight[ {}, 120, {1, -3, 4, -3, 1, 0}] (* or *) LinearRecurrence[{-2, -2, -1}, {1, -3, 4}, 120] (* Harvey P. Dale, Jan 06 2016 *)

PROG

(PARI) Vec((1-x)/(1+2*x+2*x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

(PARI) {a(n) = [1, -3, 4, -3, 1, 0][n%6 + 1]}; /* Michael Somos, Aug 05 2015 */

CROSSREFS

Cf. A100047, A254745.

Sequence in context: A242803 A064460 A108481 * A254745 A111028 A201162

Adjacent sequences:  A078067 A078068 A078069 * A078071 A078072 A078073

KEYWORD

sign,easy

AUTHOR

N. J. A. Sloane, Nov 17 2002

EXTENSIONS

Chebyshev comment and related formulas from Wolfdieter Lang, Sep 10 2004

STATUS

approved

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Last modified May 25 04:50 EDT 2019. Contains 323539 sequences. (Running on oeis4.)