A094432 a(n) = rightmost term in M^n * [1 0 0]. M = the 3 X 3 stiffness matrix [1 -1 0 / -1 4 -3 / 0 -3 3].

0, 3, 24, 165, 1104, 7347, 48840, 324597, 2157216, 14336355, 95275896, 633179973, 4207956720, 27965034003, 185848661544, 1235103986325, 8208193936704, 54549615616707, 362523179503320, 2409238895476197, 16011202548279696

Offset: 1

Author

Gary W. Adamson, May 02 2004

Keywords

• nonn
• easy

Comments

A094431(n) = left term in M^n * [1 0 0]. A stiffness matrix in Hooke's Law governs the force on nodes of stretched or compressed springs (refer to A094431). a(n)/a(n-1) tends to 4 + sqrt(7) = 6.6457513...; a(n)/A094431(n) tends to 2 + sqrt(7). A stiffness matrix is symmetric.

Examples

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

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-9).

Crossrefs

Cf. A094431, A154245.

Programs

Mathematica

Table[(MatrixPower[{{1, -1, 0}, {-1, 4, -3}, {0, -3, 3}}, n].{1, 0, 0})[[3]], {n, 21}] (* Robert G. Wilson v, May 08 2004 *)

Formula

a(n) = (3/(2*sqrt(7)))*((4+sqrt(7))^(n-1)-(4-sqrt(7))^(n-1)). For n>1, a(n) = 3*A154245(n-1). - Francesco Daddi, Aug 02 2011

G.f.: 3*x^2/(1-8*x+9*x^2). - Bruno Berselli, Aug 03 2011

Extensions

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