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Search: id:A122773
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| A122773 |
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Triangular array from Bonacci type "field": matrices: 2*(I+A[i,j]^(-1))^(-1)=Sum[A[i,j]^n,{n,0,Infinity}] Characteristic Polynomials: 2, 1 - x, -4 + 2 x + x^2, 4 - 4 x + 2 x^2 - x^3, -16 + 24 x - 16 x^2 + 4 x^3 + x^4, 16 - 32 x + 28 x^2 - 12 x^3 + 3 x^4 - x^5. |
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2
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| 2, 1, -1, -4, 2, 1, 4, -4, 2, -1, -16, 24, -16, 4, 1, 16, -32, 28, -12, 3, -1, -64, 160, -176, 104, -36, 6, 1, 64, -192, 256, -192, 88, -24, 4, -1, -256, 896, -1408, 1280, -736, 272, -64, 8, 1, 256, -1024, 1856, -1984, 1376, -640, 200, -40, 5, -1, -1024, 4608, -9472, 11648, -9472, 5312, -2080, 560, -100, 10, 1, 1024
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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It is necessary to multiply by 2 to get a repeating 1/2 factor out. 2 X 2: {{-2, 2}, {2, 0}} 3 X 3: {{1, 2, -1}, {-1, 0, 1}, {1, 0, 1}} 4 X 4: {{-2, 2, -4, 2}, {2, 0, 4, -2}, {-2, 0, -2, 2}, {2, 0, 2, 0}}, 5 X 5: {{1, 2, -1, 2, -1}, {-1, 0, 1, -2, 1}, {1, 0, 1, 2, -1}, {-1, 0, -1, 0, 1}, {1, 0, 1, 0, 1}}, 6 X 6: {{-2, 2, -4, 2, -4, 2}, {2, 0, 4, -2, 4, -2}, {-2, 0, -2, 2, -4, 2}, {2, 0, 2, 0, 4, -2}, {-2, 0, -2, 0, -2, 2}, {2, 0, 2, 0, 2, 0}}
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REFERENCES
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Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.
Kappraff, J., Blackmore, D., and Adamson, G. "Phyllotaxis as a Dynamical System: A Study in Number." In Symmetry in Plants edited by R.V. Jean and D. Barabe. Singapore: World Scientific. (1996).
Peter Steinbach, "Golden Fields: A Case for the Heptagon", Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
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FORMULA
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a(i,j)=(I[n]+M(i,j)^(-1))^(-1) 2*a(i,j)->p(m,x) p(n,x)->t(n,m)
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EXAMPLE
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Triangular array:
{2},
{1, -1},
{-4, 2, 1},
{4, -4, 2, -1},
{-16, 24, -16, 4, 1},
{16, -32, 28, -12, 3, -1},
{-64, 160, -176, 104, -36, 6, 1},
{64, -192, 256, -192, 88, -24, 4, -1}
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MATHEMATICA
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An[d_] := Table[If[n == d, 1, If[m == n + 1, 1, 0]], {n, 1, d}, {m, 1, d}]; Join[{{2}}, Table[CoefficientList[CharacteristicPolynomial[2*IdentityMatrix[d] + MatrixPower[An[d], -1], x], x], {d, 1, 20}]] Flatten[%]
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CROSSREFS
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Adjacent sequences: A122770 A122771 A122772 this_sequence A122774 A122775 A122776
Sequence in context: A098050 A111579 A144018 this_sequence A029268 A064191 A127420
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KEYWORD
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uned,tabl,sign
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AUTHOR
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Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 20 2006
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