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A094431 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].

%I A094431 #29 Dec 27 2024 16:35:08
%S A094431 1,1,2,7,38,241,1586,10519,69878,464353,3085922,20508199,136292294,
%T A094431 905764561,6019485842,40004005687,265856672918,1766817332161,
%U A094431 11741828601026,78033272818759,518589725140838,3446418345757873,22904039239795442,152214548806542679,1011580037294182454,6722709359094575521
%N A094431 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 A094431 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 A094431 Carl D. Meyer, "Matrix Analysis and Applied Linear Algebra" SIAM, 2000, p. 86.
%H A094431 Andrew Howroyd, <a href="/A094431/b094431.txt">Table of n, a(n) for n = 0..500</a>
%H A094431 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-9).
%F A094431 From _Colin Barker_, Apr 02 2012: (Start)
%F A094431 a(n) = 8*a(n-1) - 9*a(n-2).
%F A094431 G.f.: (1 - 7*x + 3*x^2)/(1 - 8*x + 9*x^2). (End)
%e A094431 a(4) = 38 since M^4 * [1 0 0] =[38 -203 165].
%t A094431 Table[(MatrixPower[{{1, -1, 0}, {-1, 4, -3}, {0, -3, 3}}, n].{1, 0, 0})[[1]], {n, 21}] (* _Robert G. Wilson v_ *)
%Y A094431 Cf. A094432.
%K A094431 nonn,easy
%O A094431 0,3
%A A094431 _Gary W. Adamson_, May 02 2004
%E A094431 More terms from _Robert G. Wilson v_, May 08 2004
%E A094431 a(0)=1 prepended by _Andrew Howroyd_, Dec 27 2024