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Search: id:A094248
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| A094248 |
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Consider 3 X 3 matrix M = [0 1 0 / 0 0 1 / 5 2 0]; a(n) = the center term in M^n * [1 1 1]. |
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| 1, 7, 7, 19, 49, 73, 193, 391, 751, 1747, 3457, 7249, 15649, 31783, 67543, 141811, 294001, 621337, 1297057, 2712679, 5700799, 11910643, 24964993, 52325281, 109483201, 229475527, 480592807, 1006367059
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OFFSET
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1,2
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COMMENT
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A sequence generated from a polynomial explored by Newton.
Barbeau quotes Isaac Newton's "Analysis by Equations of an Infinite Number of Terms", providing Newton's "Method" of finding the real root of x^3 - 2x - 5, in which Newton states "Finally, subducting the negative Part of the Quotient from the affirmative, I have 2.0945514... the Quotient sought".
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REFERENCES
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E. J. Barbeau, "Polynomials", Springer-Verlag, 1989, p. 170, E.43: "Newton's Method According to Newton".
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FORMULA
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Given x^3 - 2x - 5, the real root (and convergent of the sequence), 2.0945514815... is an eigenvalue of the 3 X 3 matrix M.
a(n)/a(n-1) tends to 2.0945514...; e.g. a(12)/a(11) = 7249/3457 = 2.0969048...
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EXAMPLE
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a(5) = 49, the center term in M^n * [1 1 1] which = [ 19 49 73].
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CROSSREFS
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Sequence in context: A070919 A070847 A053416 this_sequence A062368 A059772 A114369
Adjacent sequences: A094245 A094246 A094247 this_sequence A094249 A094250 A094251
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KEYWORD
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nonn
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AUTHOR
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Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 24 2004
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EXTENSIONS
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Corrected by T. D. Noe (noe(AT)sspectra.com), Nov 07 2006
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