

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
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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



