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A094431
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a(n) = left term in M^n * [1 0 0], where M = the 3 X 3 matrix [1 -1 0 / -1 4 -3 / 0 -3 3].
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3
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1, 2, 7, 38, 241, 1586, 10519, 69878, 464353, 3085922, 20508199, 136292294, 905764561, 6019485842, 40004005687, 265856672918, 1766817332161, 11741828601026, 78033272818759, 518589725140838, 3446418345757873
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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 4 + sqrt(7) = 6.6457513... A094432(n)/a(n) tends to 2 + sqrt(7) = 4.645638... 3. M is a "stiffness matrix" K = [k1 -k1 0 / -k1 (k1 + k2) -k2 / 0 -k2 k2] with k1 = 1, k2 = 3. K governs the force exerted on a spring with nodes, in comparison with the spring in a "no tension" position (Fig 3.2.1, p. 86, Meyer). "Stretching or compressing the springs creates a force on each node according to Hooke's law that says that the force exerted by a spring is F = kx where x is the distance the spring is stretched or compressed and where k is the stiffness constant inherent to the spring".
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REFERENCES
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Carl D. Meyer, "Matrix Analysis and Applied Linear Algebra" SIAM, 2000, p. 86.
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LINKS
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FORMULA
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Conjecture: a(n) = 8*a(n-1)-9*a(n-2). G.f.: x*(1-6*x)/(1-8*x+9*x^2). [Colin Barker, Apr 02 2012]
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EXAMPLE
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a(4) = 38 since M^4 * [1 0 0] =[38 -203 165].
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MATHEMATICA
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Table[(MatrixPower[{{1, -1, 0}, {-1, 4, -3}, {0, -3, 3}}, n].{1, 0, 0})[[1]], {n, 21}] (* Robert G. Wilson v *)
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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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