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A095128
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a(n+3) = 3*a(n+2) + 2*a(n+1) - a(n).
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4
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1, 4, 13, 46, 160, 559, 1951, 6811, 23776, 82999, 289738, 1011436, 3530785, 12325489, 43026601, 150199996, 524327701, 1830356494, 6389524888, 22304959951, 77863573135, 271811114419, 948855529576, 3312325244431, 11562875678026, 40364421993364, 140906692091713
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OFFSET
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1,2
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COMMENTS
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A sequence generated from an inverse Bell matrix, M.
a(n)/a(n-1) tends to 3.4908636153..., which is a root of x^3 - 3*x^2 - 2*x + 1 and an eigenvalue of M. A095127 is generated from the reflected polynomial: x^3 - 2*x^2 - 3*x + 1 and the inverse matrix of M. Bell numbers are sums of row terms of the 3rd-order Stirling number of the second kind matrix shown on p. 171 of Aldrovandi, the matrix being [1 0 0 / 1 1 0 / 1 3 1]. Rotations, or inverses, or related polynomials generate A095125, A095126, A095127, A095128.
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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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LINKS
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FORMULA
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Invert the matrix used to generate A095127, getting M = [3 2 -1 / 1 0 0 / 0 1 0]. Then M^n * [1 1 1] = [p q r] where a(n) = the center term q.
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EXAMPLE
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a(6) = 559 = 3*a(5) + 2*a(4) - a(3) = 3*160 + 2*46 - 13.
a(4) = 46 since M^4 * [1 1 1] = [160 46 13] = [a(5) a(4) a(3)].
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MATHEMATICA
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a[n_] := (MatrixPower[{{3, 2, -1}, {1, 0, 0}, {0, 1, 0}}, n].{{1}, {1}, {1}})[[2, 1]]; Table[ a[n], {n, 24}] (* Robert G. Wilson v, Jun 01 2004 *)
LinearRecurrence[{3, 2, -1}, {1, 4, 13}, 30] (* Harvey P. Dale, Dec 14 2012 *)
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PROG
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(Magma) I:=[1, 4, 13]; [n le 3 select I[n] else 3*Self(n-1)+2*Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jul 25 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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