A123343 Polygon cycle matrices as their characteristic polynomials to form a triangular array.

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

Offset: 1

Author

Gary W. Adamson, Oct 11 2006

Keywords

• sign
• tabl

Comments

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

Examples

			{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}}
		

Links

• Eric Weisstein's Mathworld: Adjacency Matrix
• Eric Weisstein's Mathworld: Characteristic Polynomial
• Eric Weisstein's World of Mathematics, Cycle Graph

Crossrefs

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

Cf. A049310.

Programs

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 *)

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