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].

1, 1, 2, 7, 38, 241, 1586, 10519, 69878, 464353, 3085922, 20508199, 136292294, 905764561, 6019485842, 40004005687, 265856672918, 1766817332161, 11741828601026, 78033272818759, 518589725140838, 3446418345757873, 22904039239795442, 152214548806542679, 1011580037294182454, 6722709359094575521

Offset: 0

Author

Gary W. Adamson, May 02 2004

Keywords

• nonn
• easy

Comments

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".

Examples

			a(4) = 38 since M^4 * [1 0 0] =[38 -203 165].
		

References

• Carl D. Meyer, "Matrix Analysis and Applied Linear Algebra" SIAM, 2000, p. 86.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (8,-9).

Crossrefs

Cf. A094432.

Programs

Mathematica

Table[(MatrixPower[{{1, -1, 0}, {-1, 4, -3}, {0, -3, 3}}, n].{1, 0, 0})[[1]], {n, 21}] (* Robert G. Wilson v *)

Formula

From Colin Barker, Apr 02 2012: (Start)

a(n) = 8*a(n-1) - 9*a(n-2).

G.f.: (1 - 7*x + 3*x^2)/(1 - 8*x + 9*x^2). (End)

Extensions

More terms from Robert G. Wilson v, May 08 2004

a(0)=1 prepended by Andrew Howroyd, Dec 27 2024