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A198637 Coefficient table for the characteristic polynomials of the adjacency matrices of the cycle graphs C_n. 2
1, 0, 1, -4, 0, 1, -2, -3, 0, 1, 0, 0, -4, 0, 1, -2, 5, 0, -5, 0, 1, -4, 0, 9, 0, -6, 0, 1, -2, -7, 0, 14, 0, -7, 0, 1, 0, 0, -16, 0, 20, 0, -8, 0, 1, -2, 9, 0, -30, 0, 27, 0, -9, 0, 1, -4, 0, 25, 0, -50, 0, 35, 0, -10, 0, 1, -2, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, 0, 0, -36, 0, 105, 0, -112, 0, 54, 0, -12, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
The proof for the row polynomials C(n,x), n>=2, follows by repeated expansion of the determinant, using the Chebyshev S-polynomials recurrence. For n=0 one defines C(0,x):=1, and for n=1 one has C(1,x)=x.
Modulo signs and first terms, essentially the same as A123343. - Eric W. Weisstein, Apr 05 2017
LINKS
Eric Weisstein's Mathworld: Adjacency Matrix
Eric Weisstein's Mathworld: Characteristic Polynomial
Eric Weisstein's Mathworld: Cycle Graph
FORMULA
a(n,m)=[x^m]C(n,x), with C(0,x):=1, C(1,x)=x, and
C(n,x) = 2*(T(n,x/2)-1) = R(n,x)-2 , for n>=2, with Chebyshev's T-polynomial or its monic integer version R(n,x) (usually called Chebyshev C-polynomials) with coefficient table A127672, from which a formula for a(n,m) follows. Only the column m=0 differs.
EXAMPLE
The table begins
n\m 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: 0 1
2: -4 0 1
3: -2 -3 0 1
4: 0 0 -4 0 1
5: -2 5 0 -5 0 1
6: -4 0 9 0 -6 0 1
7: -2 -7 0 14 0 -7 0 1
8: 0 0 -16 0 20 0 -8 0 1
9: -2 9 0 -30 0 27 0 -9 0 1
10: -4 0 25 0 -50 0 35 0 -10 0 1
...
C(4,x) = -4*x^2 - x^4, with zeros 2, 0, -2, 0.
C(5,x) =-2 + 5*x - 5*x^3 + x^5, with zeros 2, phi-1, -phi, -phi and phi-1, with the golden section phi:=(1+sqrt(5))/2.
The adjacency matrix for C_1 is [[0]],
for C_2 it is [[0,2],[2,0]], and for C_3 it is [[0,1,1],[1,0,1],[1,1,0]].
MATHEMATICA
Flatten[{{1}, {0, 1}, Table[(-1)^n CoefficientList[CharacteristicPolynomial[AdjacencyMatrix[CycleGraph[n]], x], x], {n, 2, 10}]}] (* Eric W. Weisstein, Apr 05 2017 *)
Flatten[{{1}, {0, 1}, Table[CoefficientList[2 (ChebyshevT[n, x/2] - 1), x], {n, 2, 10}]}] (* Eric W. Weisstein, Apr 05 2017 *)
CROSSREFS
Cf. A127672.
Cf. A123343 (essentially the same sequence).
Sequence in context: A126222 A071637 A141277 * A365094 A179296 A196817
KEYWORD
sign,easy,tabl
AUTHOR
Wolfdieter Lang, Nov 08 2011
STATUS
approved

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Last modified April 24 15:18 EDT 2024. Contains 371960 sequences. (Running on oeis4.)