A094432 - OEIS

%I #21 Jun 13 2015 00:51:20

%S 0,3,24,165,1104,7347,48840,324597,2157216,14336355,95275896,

%T 633179973,4207956720,27965034003,185848661544,1235103986325,

%U 8208193936704,54549615616707,362523179503320,2409238895476197,16011202548279696

%N 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].

%C 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.

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

%H Vincenzo Librandi, <a href="/A094432/b094432.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-9).

%F 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

%F G.f.: 3*x^2/(1-8*x+9*x^2). - _Bruno Berselli_, Aug 03 2011

%e a(4) = 165 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})[[3]], {n, 21}] (* _Robert G. Wilson v_, May 08 2004 *)

%Y Cf. A094431, A154245.

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, May 02 2004

%E More terms from _Robert G. Wilson v_, May 08 2004