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A095002
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a(n+3) = 9*a(n+2) - 9*a(n+1) + a(n); given a(1) = 1, a(2) = 3, a(3) = 19.
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3
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1, 3, 19, 145, 1137, 8947, 70435, 554529, 4365793, 34371811, 270608691, 2130497713, 16773373009, 132056486355, 1039678517827, 8185371656257, 64443294732225, 507360986201539, 3994444594880083, 31448195772839121
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| A companion to A095003, A005004; a(n)/a(n-1) tending to 4 + sqrt(15).
a(n)/a(n-1) tends to C = 4 + sqrt(15); C having the property that C + 1/C = 8. Eigenvalues of M (1, C, 1/C) are roots to x^3 - 9x^2 + 9x - 1.
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FORMULA
| Let M = a 3 X 3 symmetric Pascal's triangle matrix [1 1 1 / 1 2 3 / 1 3 6]. M^n * [1 0 0] = [A095002(a) A095003(a) A095004(a)].
O.g.f.: x(1-6x+x^2)/((1-x)(1-8x+x^2)). a(n)= (2+A001090(n+1)-7*A001090(n))/3. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 22 2008]
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EXAMPLE
| a(4) = 145 = 9*19 - 9*3 + 1.
a(4) = 145, leftmost term in M^4 * [1 0 0 = [145 352 640].
a[n_] := (MatrixPower[{{1, 1, 1}, {1, 2, 3}, {1, 3, 6}}, n].{{1}, {0},
{0}})[[1, 1]]; Table[ a[n], {n, 20}]; (from Robert G. Wilson v May 29 2004)
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CROSSREFS
| Cf. A095003, A095004, A076765.
Sequence in context: A082758 A110525 A058859 * A080833 A073516 A005258
Adjacent sequences: A094999 A095000 A095001 * A095003 A095004 A095005
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KEYWORD
| nonn
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), May 27 2004
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EXTENSIONS
| Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), May 29 2004
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