|
%I #16 May 26 2017 16:05:20
%S 1,2,7,38,241,1586,10519,69878,464353,3085922,20508199,136292294,
%T 905764561,6019485842,40004005687,265856672918,1766817332161,
%U 11741828601026,78033272818759,518589725140838,3446418345757873
%N 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].
%C 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".
%D Carl D. Meyer, "Matrix Analysis and Applied Linear Algebra" SIAM, 2000, p. 86.
%F 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]
%e a(4) = 38 since M^4 * [1 0 0] =[38 -203 165].
%t Table[(MatrixPower[{{1, -1, 0}, {-1, 4, -3}, {0, -3, 3}}, n].{1, 0, 0})[[1]], {n, 21}] (* _Robert G. Wilson v_ *)
%Y Cf. A094432.
%K nonn
%O 1,2
%A _Gary W. Adamson_, May 02 2004
%E More terms from _Robert G. Wilson v_, May 08 2004
|
