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Search: id:A123343
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| A123343 |
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Polygon cycle matrices as their characteristic polynomials to form a triangular array. |
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+0
4
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| 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; listen)
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OFFSET
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1,7
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LINKS
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Eric Weisstein's World of Mathematics, Cycle Graph
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FORMULA
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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
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EXAMPLE
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{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}}
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A049310.
Sequence in context: A156439 A087734 A073644 this_sequence A054439 A064722 A123735
Adjacent sequences: A123340 A123341 A123342 this_sequence A123344 A123345 A123346
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KEYWORD
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sign,tabl
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AUTHOR
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Gary Adamson (qntmpkt(AT)yahoo.com), Oct 11 2006
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