A176742 Expansion of (1 - x^2) / (1 + x^2) in powers of x.

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

Offset: 0

Author

Wolfdieter Lang, Oct 15 2010

Keywords

• sign
• easy

Comments

Difference sequence of A057077.

Sequence of determinants of matrices for some bipartite graphs, called Tz(n). The graph Tz(4) appears in the logo for the beer called Tannenzäpfle (small fir cone), brewed by Badische Staatsbrauerei Rothaus, Germany, hence the name Tz. See the link for this logo with Tz(4).

The vertex-vertex matrix for these bipartite graphs will also be called Tz(n) (without leading to confusion).

General proof by expanding the determinant a(n) = determinant(Tz(n)) along the first column yielding b(n-1)-b(n-2), with b(n-1) the A_{1,1} minor of the matrix Tz(n), and deriving a recurrence for the b(n), namely b(n) = -b(n-2) with inputs b(0) = 1 = b(1). This gives b(n) = A057077(n), n>=0.

Examples

			G.f. = 1 - 2*x^2 + 2*x^4 - 2*x^6 + 2*x^8 - 2*x^10 + 2*x^12 - 2*x^14 + 2*x^16 + ...
The bipartite graphs Tz(n) (n>=1) look like |, |X|, |XX|, |XXX|, ... For n>=2 the lines have to be connected to give the 2*n nodes and 2*n edges. The n=1 graph Tz(1) has 2*1=2 nodes and only one edge.
n=1: determinant((1))=1; n=2: determinant(Matrix([[1,1],[1,1]]))=0; n=3: determinant(Matrix([[1,1,0],[1,0,1],[0,1,1]]))=-2; n=4: determinant(Tz(4))=0; etc.
		

Links

• Wolfdieter Lang, Some Tz(n) graphs and matrices.
• Rothaus Tannenzäpfle, Tz(4) graph on the Tannenzäpfle logo
• Index entries for linear recurrences with constant coefficients, signature (0,-1).

Crossrefs

Cf. A084099, A057077.

Programs

Maple

a := n -> 2^signum(n)*(-1)^iquo(n+1,2)*modp(n+1,2);
seq(a(n), n=0..100); # Peter Luschny, Jun 22 2014

Mathematica

Join[{1}, Table[{0, -2, 0, 2}, {26}]] // Flatten (* Jean-François Alcover, Jun 21 2013 *)
a[ n_] := - Boole[n == 0] + {0, -2, 0, 2}[[Mod[ n, 4, 1]]]; (* Michael Somos, May 05 2015 *)
PadRight[{1},120,{2,0,-2,0}] (* Harvey P. Dale, Apr 13 2019 *)

PARI

{a(n) = - (n == 0) + [2, 0, -2, 0][n%4 + 1]}; /* Michael Somos, Mar 21 2011 */

Python

def A176742(n): return (2,0,-2,0)[n&3] if n else 1 # Chai Wah Wu, Apr 22 2025

Formula

Euler transform of length 4 sequence [0, -2, 0, 1]. - Michael Somos, Mar 21 2011

Moebius transform is length 4 sequence [0, -2, 0, 4]. - Michael Somos, Mar 22 2011

a(n) = a(-n) for all n in Z. a(n) = c_4(n) if n>1, where c_k(n) is Ramanujan's sum. - Michael Somos, Mar 21 2011

a(n-1) := determinant(Tz(n)), n>=1. The rows of the matrix Tz(4) are [[1, 1, 0, 0], [1, 0, 1, 0], [0, 1, 0, 1], [0, 0, 1, 1]]. Tz(1)=(1), and Tz(2) has rows [[1, 1], [1, 1]]. The matrix for the generalization Tz(n) has rows [[1,1,0,...,0], [1,0,1,0,...,0], [0,1,0,1,0,...,0], ..., [0,...,0,1,0,1], [0,...,0,1,1].

a(0)=1, a(2*k-1)= 0, a(4*k) = +2, a(4*k-2) = -2, k>=1.

O.g.f.: (1-x^2)/(1+x^2).

a(n) = A057077(n) - A057077(n-1), n>=1. a(0)=1.

Dirichlet g.f. sum_{n>=1} a(n)/n^s = zeta(s)*(4^(1-s)-2^(1-s)). - R. J. Mathar, Apr 11 2011

a(n) = (((n+1) mod 2)+((n+2+sign(n)) mod 2))*(-1)^ceiling(n/2). - Wesley Ivan Hurt, Jun 20 2014