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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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 A181652

Adjacent sequences:  A136702 A136703 A136704 * A136706 A136707 A136708

KEYWORD

tabl,easy,sign

AUTHOR

Roger L. Bagula, Apr 06 2008

STATUS

approved

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Last modified July 18 11:25 EDT 2019. Contains 325138 sequences. (Running on oeis4.)