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A123343 Polygon cycle matrices as their characteristic polynomials to form a triangular array. 4
1, 1, -1, -1, 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, 2, -13, 0, 91, 0, -182, 0, 156, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Modulo signs and first terms, essentially the same as A198637. - Eric W. Weisstein, Apr 05 2017

LINKS

Table of n, a(n) for n=1..100.

Eric Weisstein's Mathworld: Adjacency Matrix

Eric Weisstein's Mathworld: Characteristic Polynomial

Eric Weisstein's World of Mathematics, Cycle Graph

FORMULA

An(d) := Table[If[ n == m + 1 || n == m - 1, 1, If[ ( n == 1 && m == d) || (n == d && m == 1), 1, 0]], {n, 1, d}, {m, 1, d}] CharacteristicPloynomial[An[d]]->d=0 to 20

EXAMPLE

{1}, ( added to complete the triangle as point matrix)

{1, -1},

{-1, 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}

Matrices are:

2 X 2:

{{0, 1},

{1, 0}}

3 X 3 ( triangle like):

{{0, 1, 1},

{1, 0, 1},

{1, 1, 0}}

4 X 4

{{0, 1, 0, 1},

{1, 0, 1, 0},

{0, 1, 0, 1},

{1, 0, 1, 0}}

5 X 5

{{0, 1, 0, 0, 1},

{1, 0, 1, 0, 0},

{0, 1, 0, 1, 0},

{0, 0, 1, 0, 1},

{1, 0, 0, 1, 0}}

MATHEMATICA

An[d_] := Table[If[ n == m + 1 || n == m - 1, 1, If[ ( n == 1 && m == d) || (n == d && m == 1), 1, 0]], {n, 1, d}, {m, 1, d}] Table[An[d], {d, 2, 20}] Table[CharacteristicPolynomial[An[d], x], {d, 2, 20}] Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[ An[d], x], x], {d, 1, 20}]] Flatten[%] Table[NSolve[CharacteristicPolynomial[An[d], x] == 0, x], {d, 2, 20}]

Flatten[{{1}, {1, -1}, {-1, 0, 1}, Table[CoefficientList[CharacteristicPolynomial[AdjacencyMatrix[CycleGraph[n]], x], x], {n, 3, 10}]}] (* Eric W. Weisstein, Apr 05 2017 *)

Flatten[{{1}, {1, -1}, {-1, 0, 1}, Table[CoefficientList[(-1)^n 2 (ChebyshevT[n, x/2] - 1), x], {n, 3, 10}]}] (* Eric W. Weisstein, Apr 05 2017 *)

CROSSREFS

Cf. A198637 (essentially the same sequence). - Eric W. Weisstein, Apr 06 2017

Cf. A049310.

Sequence in context: A156439 A087734 A073644 * A054439 A318656 A215151

Adjacent sequences:  A123340 A123341 A123342 * A123344 A123345 A123346

KEYWORD

sign,tabl

AUTHOR

Gary W. Adamson, Oct 11 2006

STATUS

approved

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Last modified August 1 19:59 EDT 2021. Contains 346402 sequences. (Running on oeis4.)