A123343 - OEIS

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 (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`