login
This site is supported by donations to The OEIS Foundation.

 

Logo

Many excellent designs for a new banner were submitted. We will use the best of them in rotation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A176742 Expansion of (1 - x^2) / (1 + x^2) in powers of x. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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 Tannenzaepfle (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.

LINKS

Table of n, a(n) for n=0..104.

Tz(4) graph on the Tannenzaepfle logo

W. Lang, Some Tz(n) graphs and matrices. [From Wolfdieter Lang, Oct 28 2010]

Index to sequences with linear recurrences with constant coefficients, signature (0,-1)

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). 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

EXAMPLE

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.

MATHEMATICA

Join[{1}, Table[{0, -2, 0, 2}, {26}]] // Flatten  (* Jean-Fran├žois Alcover, Jun 21 2013 *)

PROG

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

CROSSREFS

Cf. A084099, A057077.

Sequence in context: A101227 A230103 A021499 * A010673 A084099 A036665

Adjacent sequences:  A176739 A176740 A176741 * A176743 A176744 A176745

KEYWORD

sign,easy

AUTHOR

Wolfdieter Lang, Oct 15 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified April 23 06:07 EDT 2014. Contains 240913 sequences.