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%I #54 Jun 11 2019 07:00:36
%S 0,1,1,-1,-1,0,-1,-1,1,1,0,1,1,-1,-1,0,-1,-1,1,1,0,1,1,-1,-1,0,-1,-1,
%T 1,1,0,1,1,-1,-1,0,-1,-1,1,1,0,1,1,-1,-1,0,-1,-1,1,1,0,1,1,-1,-1,0,-1,
%U -1,1,1,0,1,1,-1,-1,0,-1,-1,1,1,0,1,1,-1,-1,0,-1,-1,1,1,0,1,1,-1,-1,0,-1,-1,1,1
%N A Chebyshev transform of the Fibonacci numbers.
%C Multiplicative with a(p^e) = (-1)^(e+1) if p = 2, 0 if p = 5, 1 if p == 1 or 9 (mod 10), (-1)^e if p == 3 or 7 (mod 10). - _David W. Wilson_, Jun 10 2005
%C This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = 1, P2 = -1, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Mar 24 2014
%C From _Peter Bala_, Mar 24 2014: (Start)
%C This is the particular case P1 = 1, P2 = -1, Q = 1 of the following results:
%C Let P1, P2 and Q be integers. Let alpha and beta denote the roots of the quadratic equation x^2 - 1/2*P1*x + 1/4*P2 = 0. Let T(n,x;Q) denote the bivariate Chebyshev polynomial of the first kind defined by T(n,x;Q) = 1/2*( (x + sqrt(x^2 - Q))^n + (x - sqrt(x^2 - Q))^n ) (when Q = 1, T(n,x;Q) reduces to the ordinary Chebyshev polynomial of the first kind T(n,x)). Then we have
%C 1) The sequence A(n) := ( T(n,alpha;Q) - T(n,beta;Q) )/(alpha - beta) is a linear divisibility sequence of the fourth order.
%C 2) A(n) belongs to the 3-parameter family of fourth-order divisibility sequences found by Williams and Guy.
%C 3) The o.g.f. of the sequence A(n) is the rational function x*(1 - Q*x^2)/(1 - P1*x + (P2 + 2*Q)*x^2 - P1*Q*x^3 + Q^2*x^4).
%C 4) The o.g.f. is the Chebyshev transform of the rational function x/(1 - P1*x + P2*x^2), where the Chebyshev transform takes the function A(x) to the function (1 - Q*x^2)/(1 + Q*x^2)*A(x/(1 + Q*x^2)).
%C 5) Let q = sqrt(Q) and set a = sqrt( q + (P2)/(4*q) + (P1)/2 ) and b = sqrt( q + (P2)/(4*q) - (P1)/2 ). Then the o.g.f. of the sequence A(n) is the Hadamard product of the rational functions x/(1 - (a + b)*x + q*x^2) and x/(1 - (a - b)*x + q*x^2). Thus A(n) is the product of two (usually, non-integer) Lucas-type sequences.
%C 6) A(n) = the bottom left entry of the 2 X 2 matrix 2*T(n,1/2*M;Q), where M is the 2 X 2 matrix [0, -P2; 1, P1].
%C For examples of the above see A006238, A054493, A078070, A092184, A098306, A100048, A108196, A138573, A152090 and A218134. (End)
%H G. C. Greubel, <a href="/A100047/b100047.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 Tian-Xiao He, Peter J.-S. Shiue, <a href="https://www.emis.de/journals/JIS/VOL20/He/he59.html">An Approach to the Construction of Linear Divisibility Sequences of Higher Orders</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
%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_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,-1).
%F G.f.: x*(1 - x^2)/(1 - x + x^2 - x^3 + x^4).
%F a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4).
%F a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k *binomial(n-k, k)*A000045(n-2*k)/(n -k).
%F From _Peter Bala_, Mar 24 2014: (Start)
%F a(n) = (T(n,alpha) - T(n,beta))/(alpha - beta), where alpha = (1 + sqrt(5))/4 and beta = (1 - sqrt(5))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.
%F a(n) = bottom left entry of the matrix T(n, M), where M is the 2 X 2 matrix [0, 1/4; 1, 1/2].
%F The o.g.f. is the Hadamard product of the rational functions x/(1 - 1/2*(sqrt(5) + 1)*x + x^2) and x/(1 - 1/2*(sqrt(5) - 1)*x + x^2). (End)
%F Euler transform of length 10 sequence [ 1, -2, 0, 0, -1, 0, 0, 0, 0, 1]. - _Michael Somos_, May 24 2015
%F a(n) = a(-n) = -a(n + 5) for all n in Z. - _Michael Somos_, May 24 2015
%F |A011558(n)| = |A080891(n)| = |a(n)| = A244895(n). - _Michael Somos_, May 24 2015
%e A Chebyshev transform of the Fibonacci numbers A000045: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).
%e The denominator is the 10th cyclotomic polynomial.
%e G.f. = x + x^2 - x^3 - x^4 - x^6 - x^7 + x^8 + x^9 + x^11 + x^12 - x^13 + ...
%p a := n -> (-1)^iquo(n, 5)*signum(mods(n, 5)):
%p seq(a(n), n=0..89); # after _Michael Somos_, _Peter Luschny_, Dec 30 2018
%t a[ n_] := {1, 1, -1, -1, 0, -1, -1, 1, 1, 0}[[Mod[ n, 10, 1]]]; (* _Michael Somos_, May 24 2015 *)
%t a[ n_] := (-1)^Quotient[ n, 5] Sign[ Mod[ n, 5, -2]]; (* _Michael Somos_, May 24 2015 *)
%t a[ n_] := (-1)^Quotient[n, 5] {1, 1, -1, -1, 0}[[Mod[ n, 5, 1]]]; (* _Michael Somos_, May 24 2015 *)
%t LinearRecurrence[{1, -1, 1, -1}, {0, 1, 1, -1}, 90] (* _Jean-François Alcover_, Jun 11 2019 *)
%o (PARI) {a(n) = (-1)^(n\5) * [0, 1, 1, -1, -1][n%5+1]}; /* _Michael Somos_, May 24 2015 */
%o (PARI) {a(n) = (-1)^(n\5) * sign( centerlift( Mod(n, 5)))}; /* _Michael Somos_, May 24 2015 */
%Y Other Chebyshev transforms: A011655, A099443, A100048.
%Y Cf. A006238, A054493, A078070, A092184, A098306, A108196, A138573, A152090, A218134.
%Y Cf. also A011558, A080891, A244895, A226162.
%K easy,sign,mult
%O 0,1
%A _Paul Barry_, Oct 31 2004
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