A136705 Triangle read by rows where the n-th row gives the coefficients of the characteristic polynomial for a Fibonacci-type matrix with a=1 and b=1.

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

Offset

0,1

Comments

Row sums are 1, 0, -1, 1, -1, 1, -1, 1, -1, 1, ... .

Links

Table of n, a(n) for n=0..73.

J. Cigler, q-Fibonacci polynomials, Fibonacci Quarterly 41 (2003) 31-40.

Formula

The n-th row contains the coefficients (from lowest-order to highest-order) of the characteristic polynomial of the matrix with (i,j)-entry given by: if(i = j = n, 1, if(j = n and i = 1, 1, if(i = j + 1, 1, 0))).

For n >= 2, the n-th row of the triangle consists of (-1)^(n+1), followed by n-2 zeros, followed by (-1)^(n+1) and (-1)^n. - Nathaniel Johnston, Apr 27 2011

Example

Triangle begins:
1,
1, -1,
-1, -1, 1,
1, 0, 1, -1,
-1, 0, 0, -1, 1,
1, 0, 0, 0, 1, -1,
-1, 0, 0, 0, 0, -1, 1,
1, 0, 0, 0, 0, 0, 1, -1,
-1, 0, 0, 0, 0, 0, 0, -1, 1,
1, 0, 0, 0, 0, 0, 0, 0, 1, -1,
-1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1,
...
For n = 4, the matrix is {{0, 0, 0, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 1}}.

Mathematica

T[n_, m_, d_] := If[ n == m == d, 1, If[m == d && n == 1, 1, If[n == m + 1, 1, 0]]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; Table[Det[M[d]], {d, 1, 10}]; Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]; a = Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a]

Crossrefs

The triangle when reversed is very similar to A141679. - N. J. A. Sloane, Dec 14 2014

Sequence in context: A190207 A156706 A075743 * A141646 A129573 A330887

Adjacent sequences: A136702 A136703 A136704 * A136706 A136707 A136708

Keyword

tabl,easy,sign

Author

Roger L. Bagula, Apr 06 2008

Status

approved