A100047 A Chebyshev transform of the Fibonacci numbers.

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, 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, 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

Offset: 0

Author

Paul Barry, Oct 31 2004

Keywords

• easy
• sign
• mult

Comments

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

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

From Peter Bala, Mar 24 2014: (Start)

This is the particular case P1 = 1, P2 = -1, Q = 1 of the following results:

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

1) The sequence A(n) := ( T(n,alpha;Q) - T(n,beta;Q) )/(alpha - beta) is a linear divisibility sequence of the fourth order.

2) A(n) belongs to the 3-parameter family of fourth-order divisibility sequences found by Williams and Guy.

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).

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)).

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.

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].

For examples of the above see A006238, A054493, A078070, A092184, A098306, A100048, A108196, A138573, A152090 and A218134. (End)

Examples

			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)).
The denominator is the 10th cyclotomic polynomial.
G.f. = x + x^2 - x^3 - x^4 - x^6 - x^7 + x^8 + x^9 + x^11 + x^12 - x^13 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Peter Bala, Linear divisibility sequences and Chebyshev polynomials
• Tian-Xiao He, Peter J.-S. Shiue, An Approach to the Construction of Linear Divisibility Sequences of Higher Orders, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
• 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 (1,-1,1,-1).

Crossrefs

Other Chebyshev transforms: A011655, A099443, A100048.

Cf. A006238, A054493, A078070, A092184, A098306, A108196, A138573, A152090, A218134.

Cf. also A011558, A080891, A244895, A226162.

Programs

Maple

a := n -> (-1)^iquo(n, 5)*signum(mods(n, 5)):
seq(a(n), n=0..89); # after Michael Somos, Peter Luschny, Dec 30 2018

Mathematica

a[ n_] := {1, 1, -1, -1, 0, -1, -1, 1, 1, 0}[[Mod[ n, 10, 1]]]; (* Michael Somos, May 24 2015 *)
a[ n_] := (-1)^Quotient[ n, 5] Sign[ Mod[ n, 5, -2]]; (* Michael Somos, May 24 2015 *)
a[ n_] := (-1)^Quotient[n, 5] {1, 1, -1, -1, 0}[[Mod[ n, 5, 1]]]; (* Michael Somos, May 24 2015 *)
LinearRecurrence[{1, -1, 1, -1}, {0, 1, 1, -1}, 90] (* Jean-François Alcover, Jun 11 2019 *)

PARI

{a(n) = (-1)^(n\5) * [0, 1, 1, -1, -1][n%5+1]}; /* Michael Somos, May 24 2015 */

PARI

{a(n) = (-1)^(n\5) * sign( centerlift( Mod(n, 5)))}; /* Michael Somos, May 24 2015 */

Formula

G.f.: x*(1 - x^2)/(1 - x + x^2 - x^3 + x^4).

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

a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k *binomial(n-k, k)*A000045(n-2*k)/(n -k).

From Peter Bala, Mar 24 2014: (Start)

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.

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].

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)

Euler transform of length 10 sequence [ 1, -2, 0, 0, -1, 0, 0, 0, 0, 1]. - Michael Somos, May 24 2015

a(n) = a(-n) = -a(n + 5) for all n in Z. - Michael Somos, May 24 2015

|A011558(n)| = |A080891(n)| = |a(n)| = A244895(n). - Michael Somos, May 24 2015