A123343 - OEIS

%I #11 Apr 06 2017 15:16:43

%S 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,

%T 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,

%U 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

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

%C Modulo signs and first terms, essentially the same as A198637. - _Eric W. Weisstein_, Apr 05 2017

%H Eric Weisstein's Mathworld: <a href="http://mathworld.wolfram.com/AdjacencyMatrix.html">Adjacency Matrix</a>

%H Eric Weisstein's Mathworld: <a href="http://mathworld.wolfram.com/CharacteristicPolynomial.html">Characteristic Polynomial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>

%F 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

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

%e {1, -1},

%e {-1, 0, 1},

%e {2, 3, 0, -1},

%e {0, 0, -4, 0, 1},

%e {2, -5, 0, 5, 0, -1},

%e {-4, 0, 9, 0, -6, 0, 1},

%e {2, 7, 0, -14, 0, 7,0, -1},

%e {0, 0, -16, 0, 20, 0, -8, 0, 1},

%e {2, -9, 0, 30, 0, -27,0, 9, 0, -1},

%e {-4, 0, 25, 0, -50, 0, 35, 0, -10, 0, 1},

%e {2, 11, 0, -55, 0, 77, 0, -44, 0, 11, 0, -1}

%e Matrices are:

%e 2 X 2:

%e {{0, 1},

%e {1, 0}}

%e 3 X 3 ( triangle like):

%e {{0, 1, 1},

%e {1, 0, 1},

%e {1, 1, 0}}

%e 4 X 4

%e {{0, 1, 0, 1},

%e {1, 0, 1, 0},

%e {0, 1, 0, 1},

%e {1, 0, 1, 0}}

%e 5 X 5

%e {{0, 1, 0, 0, 1},

%e {1, 0, 1, 0, 0},

%e {0, 1, 0, 1, 0},

%e {0, 0, 1, 0, 1},

%e {1, 0, 0, 1, 0}}

%t 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}]

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

%t 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 *)

%Y Cf. A198637 (essentially the same sequence). - _Eric W. Weisstein_, Apr 06 2017

%Y Cf. A049310.

%K sign,tabl

%O 1,7

%A _Gary W. Adamson_, Oct 11 2006