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A123951
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A polynomial of matrices is used to make a triangular sequence. The upper triangular anti-diagonal Steinbach matrices are summed over their characteristic polynomial triangular sequences to give a new sequence of matrices: the characteristic polynomials of these new summed matrices are, then, used to make up this triangular sequence.
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1, 1, -1, -1, -1, 1, -1, -3, 4, -1, 37, -88, 69, -19, 1, 10879, -14344, 6831, -1375, 99, -1, -4322473, -40529664, -17486038, 3188841, -40896, -2346, 1, -11384127259974047, -783824545942228, 1058675233347, 505084925760, -64007100, -32568519, 23164, -1, -121986767767877481129923
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| Basically everything is done twice. The determinants get very large very fast for these matrices: Table[Det[w[[d]]], {d, 1, Length[w]}] {1, -1, -1, 37, 10879, -4322473, -11384127259974047, -121986767767877481129923, -323621163456130064854374309178100414058036559, 189651898964129252384795657180434913387386019400002936829101989683}
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REFERENCES
| Peter Steinbach, "Golden Fields: A Case for the Heptagon", Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
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FORMULA
| p(n,x)=CharacteristicPolynomial(a(i,j)) p(n,x)->t(n,m) b(i,j)=Sum[t(i,j).a(j,k).{j,1,m}] p'(n,x)=CharacteristicPolynomial(b(i,j)) p'(n,x)->t'(n,m)
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EXAMPLE
| {1},
{1, -1},
{-1, -1, 1},
{-1, -3, 4, -1},
{37, -88, 69, -19,1},
{10879, -14344, 6831, -1375, 99, -1},
{-4322473, -40529664, -17486038, 3188841, -40896, -2346, 1}
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MATHEMATICA
| An[d_] := Table[If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; w = Join[{{{1}}}, Table[Sum[MatrixPower[a[[n]][[m + 1]]*An[n], m - 1], {m, 0, Length[a[[n]]] - 1}], {n, 2, 10}]]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[w[[d]], x], x], {d, 1, Length[w]}]]; Flatten[%]
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CROSSREFS
| Cf. A122765, A122771.
Sequence in context: A162498 A134049 A157783 * A123127 A167876 A077451
Adjacent sequences: A123948 A123949 A123950 * A123952 A123953 A123954
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KEYWORD
| uned,tabl,sign
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AUTHOR
| Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Oct 26 2006
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