

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].


3



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

1,2


COMMENTS

a(n)/a(n1) 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".


REFERENCES

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


LINKS

Table of n, a(n) for n=1..21.


FORMULA

Conjecture: a(n) = 8*a(n1)9*a(n2). G.f.: x*(16*x)/(18*x+9*x^2). [Colin Barker, Apr 02 2012]


EXAMPLE

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


MATHEMATICA

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


CROSSREFS

Cf. A094432.
Sequence in context: A180269 A036432 A275621 * A256032 A209006 A168492
Adjacent sequences: A094428 A094429 A094430 * A094432 A094433 A094434


KEYWORD

nonn


AUTHOR

Gary W. Adamson, May 02 2004


EXTENSIONS

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


STATUS

approved



