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Search: id:A093406
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| A093406 |
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A sequence converging to 1 + 2^(1/4). |
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+0
2
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| 1, 3, 11, 31, 71, 145, 289, 601, 1321, 2979, 6683, 14743, 32111, 69697, 151777, 332113, 728689, 1598883, 3503627, 7668079, 16774775, 36704017, 80343361, 175916521, 385196761
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OFFSET
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1,2
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COMMENT
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a(n)/a(n-1) tends to 2.189207115... = 1 + 2^(1/4). Example: a(18)/a(17) = 1598883/728689 = 2.19419... A052101 is the series derived from analogous 3rd order operations, with lim a(n)/a(n-1) as n approaches inf. = 1 + 2^(1/3).
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REFERENCES
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E.J. Barbeau; Polynomials, Springer-Verlag NY Inc, 1989, p. 136.
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FORMULA
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We use a 4 X 4 matrix corresponding to the characteristic polynomial (x - 1)^4 - 2 = 0 = x^4 - 4x^3 + 6x^2 - 4x - 1 = 0, being [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / 1 4 -6 4]. Let the matrix = M. Perform M^n * [1, 1, 1, 1]. a(n) = the third term from the left, (the other 3 terms being offset members of the series).
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EXAMPLE
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a(4) = 31, since M^4 * [1,1,1,1] = [3, 11, 31, 71].
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CROSSREFS
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Cf. A052101.
Adjacent sequences: A093403 A093404 A093405 this_sequence A093407 A093408 A093409
Sequence in context: A119215 A057172 A071568 this_sequence A097081 A107587 A087323
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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), Mar 28 2004
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EXTENSIONS
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Corrected by T. D. Noe (noe(AT)sspectra.com), Nov 08 2006
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