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A095002
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a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); 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, 247591121587832881, 1949280776929823923
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OFFSET
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1,2
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COMMENTS
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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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LINKS
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FORMULA
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Let M be the 3 X 3 matrix [1 1 1 / 1 2 3 / 1 3 6]. M^n * [1 0 0] = [a(n) A095003(n) A095004(n)].
O.g.f.: x*(1-6x+x^2)/((1-x)*(1-8x+x^2)).
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EXAMPLE
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a(4) = 145 = 9*19 - 9*3 + 1.
a(4) = 145, leftmost term in M^4 * [1 0 0] = [145 352 640].
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MAPLE
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a:= n-> (<<1|1|1>, <1|2|3>, <1|3|6>>^n)[1$2]:
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MATHEMATICA
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a[n_] := (MatrixPower[{{1, 1, 1}, {1, 2, 3}, {1, 3, 6}}, n].{{1}, {0},
nxt[{a_, b_, c_}]:={b, c, 9c-9b+a}; NestList[nxt, {1, 3, 19}, 30][[All, 1]] (* Harvey P. Dale, Sep 02 2022 *)
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PROG
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(PARI) Vec(x*(1-6*x+x^2)/((1-x)*(1-8*x+x^2)) + O(x^20)) \\ Michel Marcus, Mar 21 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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