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A122773 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. 1
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; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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}}

REFERENCES

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.

LINKS

Table of n, a(n) for n=1..67.

FORMULA

a(i,j)=(I[n]+M(i,j)^(-1))^(-1) 2*a(i,j)->p(m,x) p(n,x)->t(n,m)

EXAMPLE

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}

MATHEMATICA

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[%]

CROSSREFS

Sequence in context: A258709 A239144 A156861 * A029268 A176452 A064191

Adjacent sequences:  A122770 A122771 A122772 * A122774 A122775 A122776

KEYWORD

uned,tabl,sign

AUTHOR

Gary W. Adamson and Roger L. Bagula, Oct 20 2006

STATUS

approved

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Last modified February 18 03:43 EST 2018. Contains 299298 sequences. (Running on oeis4.)