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Search: id:A122771
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| A122771 |
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Triangular array from Bonacci type "field: matrices: I+A[i,j]^(-1) Characteristic Polynomials: 1, 2 - x, -1 - x + x^2, 2 - 2 x + 2 x^2 - x^3, 1 - 2x + 4 x^2 - 3 x^3 + x^4, 2 - 3 x + 6 x^2 - 7 x^3 + 4 x^4 - x^5, -1- 3 x + 9 x^2 - 13 x^3 + 11x^4 - 5x^5 + x^6, 2 - 4\x + 12x^2 - 22 x^3 + 24 x^4 - 16\x^5 + 6 x^6 - x^7. |
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+0
3
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| 1, 2, -1, -1, -1, 1, 2, -2, 2, -1, -1, -2, 4, -3, 1, 2, -3, 6, -7, 4, -1, -1, -3, 9, -13, 11, -5, 1, 2, -4, 12, -22, 24, -16, 6, -1, -1, -4, 16, -34, 46, -40, 22, -7, 1, 2, -5, 20, -50, 80, -86, 62, -29, 8, -1, -1, -5, 25, -70, 130, -166, 148, -91, 37, -9, 1, 2, -6, 30, -95, 200, -296, 314, -239, 128, -46, 10, -1, -1, -6, 36, -125
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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2 X 2: {{0, 1}, {1, 1}}, 3 X 3: {{0, -1, 1}, {1, 1, 0}, {0, 1, 1}}, 4 X 4: {{0, -1, -1, 1}, {1, 1, 0, 0}, {0, 1, 1, 0}, {0, 0, 1,1}}, 5 X 5: {{0, -1, -1, -1, 1}, {1, 1, 0, 0, 0}, {0, 1, 1, 0, 0}, {0, 0, 1, 1, 0}, {0, 0, 0, 1,1}}, 6 X 6: {{0, -1, -1, -1, -1, 1}, {1, 1, 0, 0, 0, 0}, {0, 1, 1, 0, 0, 0}, {0, 0, 1, 1, 0, 0}, {0, 0, 0, 1, 1, 0}, {0, 0, 0, 0, 1, 1}}
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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) a(i,j)->p(m,x) p(n,x)->t(n,m)
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EXAMPLE
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Triangular array:
{1},
{2, -1},
{-1, -1, 1},
{2, -2, 2, -1},
{-1, -2, 4, -3, 1},
{2, -3, 6, -7,4, -1},
{-1, -3, 9, -13, 11, -5,1},
{2, -4, 12, -22, 24, -16, 6, -1},
{-1, -4, 16, -34, 46, -40, 22, -7, 1},
{2, -5, 20, -50, 80, -86, 62, -29, 8, -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[{{1}}, Table[CoefficientList[CharacteristicPolynomial[IdentityMatrix[d] + MatrixPower[An[d], -1], x], x], {d, 1, 20}]]; Flatten[%]
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CROSSREFS
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Sequence in context: A124961 A008967 A094189 this_sequence A112188 A112189 A112190
Adjacent sequences: A122768 A122769 A122770 this_sequence A122772 A122773 A122774
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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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