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Search: id:A100058
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| A100058 |
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Expansion of 1 / (1 - 3x - x^2 + 2x^3). |
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+0
4
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| 1, 3, 10, 31, 97, 302, 941, 2931, 9130, 28439, 88585, 275934, 859509, 2677291, 8339514, 25976815, 80915377, 252043918, 785093501, 2445493667, 7617486666, 23727766663, 73909799321, 230222191294, 717120839877, 2233765112283
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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a(n)/a(n-1) tends to 3.1149075414...an eigenvalue of the matrix M and a root of the characteristic polynomial x^3 - 3x^2 - x + 2.
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REFERENCES
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Boris A. Bondarenko, "Generalized Pascal Triangles and Pyramids, Their Fractals, Graphs, and Applications", Fibonacci Association, 1993, p. 27.
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FORMULA
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Recurrence: a(0) = 1, a(1) = 3, a(2) = 10; a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3).
Given Hosoya's triangle: 1; 1, 1; 2, 1, 2; considered as an upper triangular 3 X 3 matrix M: [2 1 2 / 1 1 0 / 1 0 0]; a(n) = center term in M^n * [1 0 0].
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EXAMPLE
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a(5) = 97, center term in M^5 * [1 0 0]: [205 97 66].
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MATHEMATICA
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CoefficientList[Series[1/(1 - 3x - x^2 + 2x^3), {x, 0, 25}], x] (* Or *)
Table[(MatrixPower[{{2, 1, 2}, {1, 1, 0}, {1, 0, 0}}, n].{1, 0, 0})[[2]], {n, 26}] (from Robert G. Wilson v Nov 04 2004)
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CROSSREFS
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Partial sums of A052911. Cf. A019481, A052550, A052939, A100059, A058071.
Adjacent sequences: A100055 A100056 A100057 this_sequence A100059 A100060 A100061
Sequence in context: A055217 A097472 A068094 this_sequence A002160 A114487 A017934
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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), Oct 31 2004
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EXTENSIONS
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Edited by Ralf Stephan, Nov 02 2004
Corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 04 2004
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