A094434 a(n) = rightmost term of M^n * [1 0 0], with M = the 3 X 3 matrix [1 -1 0 / -1 3 -2 / 0 -2 2].

0, 2, 12, 60, 288, 1368, 6480, 30672, 145152, 686880, 3250368, 15380928, 72783360, 344414592, 1629787392, 7712236800, 36494696448, 172694757888, 817200368640, 3867033664512, 18298999775232, 86591796664320, 409756781334528

Offset: 1

Author

Gary W. Adamson, May 02 2004

Keywords

• nonn
• easy

Comments

Left term in M^n * [1 0 0] = A094433(n). a(n)/ a(n-1) tends to 3 + sqrt(3) = 4.732050807...; e.g. a(9)/a(8) = 145152/30672 = 4.732394... 3. a(n)/ A094433(n) tends to 1 + sqrt(3); e.g. a(9)/A094433(9) = 145152/53136 = 2.731707... 4. M = a "stiffness matrix" with k1 = 1, k2 = 2, relating to Hooke's law governing the force on the nodes of compressed or stretched springs with stiffness constants k1, k2. (see A094433, A094431).

Examples

			a(4) = 60 since M^4 * [1 0 0] = [24 -84 60].
		

References

• Carl D. Meyer, "Matrix Analysis and Applied Linear Algebra", SIAM, 2000, p. 86-87.

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-6).

Crossrefs

Cf. A094431, A094431, A094433.

Programs

Mathematica

Table[(MatrixPower[{{1, -1, 0}, {-1, 3, -2}, {0, -2, 2}}, n].{1, 0, 0})[[3]], {n, 24}] (* Robert G. Wilson v *)
LinearRecurrence[{6,-6},{0,2},30] (* Harvey P. Dale, May 01 2017 *)

Formula

a(n) = 6*a(n-1)-6*a(n-2). G.f.: 2*x^2/(1-6*x+6*x^2). [Colin Barker, Sep 05 2012]

Extensions

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