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Search: id:A095125
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| A095125 |
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a(n+3) = 2*a(n+2) + 3*(n+1) - n; with a(1) = 1, a(2) = 3, a(3) = 8. |
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5
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| 1, 3, 8, 24, 69, 202, 587, 1711, 4981, 14508, 42248, 123039, 358314, 1043497, 3038897, 8849971, 25773136, 75057288, 218584013, 636566754, 1853828259, 5398772767, 15722463557, 45787417156, 133343452216, 388326692343
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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 rotated Stirling number of the second kind matrix.
a(n)/a(n-1) tends to the largest positive eigenvalue of the matrix, 2.9122291784... (a root of the characteristic polynomial x^3 - 2x^2 - 3x + 1); e.g. a(9)/a(8) = 4981/1711 = 2.91116... A095127 is generated from an inverse of M, while A095126 is generated from M.
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REFERENCES
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R. Aldrovandi, "Special Matrices of Mathematical Physics," World Scientific, 2001, Section 13.3.1 "Inverting Bell Matrices", p. 171.
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FORMULA
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M = [1 1 1 / 3 1 0 / 1 0 0], (a rotation of a Stirling number of the second kind matrix [1 0 0 / 1 1 0 / 1 3 1]; then M^n * [1 1 1] = [a(n+1), A095126(n) a(n)].
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EXAMPLE
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a(5) = 69 = 2*a(4) + 3*a(3) - a(2) = 2*24 + 3*8 - 3.
a(5) = 69 since M^5 * [1 1 1] = [202 316 69] = [a(6) A095126(a) a(5)].
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MATHEMATICA
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a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {1, 0, 0}}, n].{{1}, {1}, {1}})[[3, 1]]; Table[ a[n], {n, 25}] (from Robert G. Wilson v Jun 01 2004)
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CROSSREFS
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Cf. A095126, A095127, A095128.
Sequence in context: A056332 A091588 A018046 this_sequence A078055 A079121 A027077
Adjacent sequences: A095122 A095123 A095124 this_sequence A095126 A095127 A095128
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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), May 29 2004
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EXTENSIONS
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Edited, corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 01 2004
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