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A136705
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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.
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2
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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
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OFFSET
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0,1
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COMMENTS
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Row sums are: 1, 0, -1, 1, -1, 1, -1, 1, -1, 1, ...
If these are vector reversed and the absolute value taken, they are the same as the Galois field triangular sequence.
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LINKS
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Table of n, a(n) for n=0..73.
J. Cigler, q-Fibonacci polynomials, Fibonacci Quarterly 41 (2003) 31-40.
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FORMULA
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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 zeroes, followed by (-1)^(n+1) and (-1)^n. - Nathaniel Johnston, Apr 27 2011
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EXAMPLE
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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}}.
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MATHEMATICA
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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]
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CROSSREFS
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Sequence in context: A167686 A156706 A075743 * A141646 A129573 A181652
Adjacent sequences: A136702 A136703 A136704 * A136706 A136707 A136708
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KEYWORD
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tabl,easy,sign
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AUTHOR
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Roger L. Bagula, Apr 06 2008
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STATUS
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approved
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