|
%I #55 Mar 23 2024 08:04:43
%S 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,
%T 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,
%U 1,0,1,-3,4,-3,1,0,1,-3,4,-3,1,0,1,-3,4,-3,1,0,1,-3
%N Expansion of (1-x)/(1 + 2*x + 2*x^2 + x^3).
%C Period 6: repeat [1, -3, 4, -3, 1, 0].
%C 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 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).
%C 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
%H Vincenzo Librandi, <a href="/A078070/b078070.txt">Table of n, a(n) for n = 0..1000</a>
%H Peter Bala, <a href="/A100047/a100047.pdf">Linear divisibility sequences and Chebyshev polynomials</a>
%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H H. C. Williams and R. K. Guy, <a href="https://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-2,-2,-1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F abs(a(n)) = 2 + 2*cos(Pi*n/3 - 2*Pi/3). - _Paul Barry_, Mar 14 2004
%F Euler transform of finite sequence [-3, 1, 1]. - _Michael Somos_, Sep 17 2004
%F 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) = 2*T(n+1, -1/2) + 2(-1)^n. - _Paul Barry_, Dec 12 2004
%F From _Peter Bala_, Mar 25 2014: (Start)
%F The following formulas assume an offset of 1.
%F 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).
%F 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.
%F 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.
%F 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.
%F See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
%F a(n) = a(n+6) = a(-2-n) for all n in Z. - _Michael Somos_, Aug 05 2015
%F a(n) = (-1)^n * A254745(n). - _Michael Somos_, Jul 16 2017
%e 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 + ...
%t a[ n_] := {-3, 4, -3, 1, 0, 1}[[Mod[ n, 6, 1]]]; (* _Michael Somos_, Aug 05 2015 *)
%t 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 *)
%o (PARI) Vec((1-x)/(1+2*x+2*x^2+x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%o (PARI) {a(n) = [1, -3, 4, -3, 1, 0][n%6 + 1]}; /* _Michael Somos_, Aug 05 2015 */
%Y Cf. A100047, A254745.
%K sign,easy
%O 0,2
%A _N. J. A. Sloane_, Nov 17 2002
%E Chebyshev comment and related formulas from _Wolfdieter Lang_, Sep 10 2004
|
