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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]. 1
0, 2, 12, 60, 288, 1368, 6480, 30672, 145152, 686880, 3250368, 15380928, 72783360, 344414592, 1629787392, 7712236800, 36494696448, 172694757888, 817200368640, 3867033664512, 18298999775232, 86591796664320, 409756781334528 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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).
REFERENCES
Carl D. Meyer, "Matrix Analysis and Applied Linear Algebra", SIAM, 2000, p. 86-87.
LINKS
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]
EXAMPLE
a(4) = 60 since M^4 * [1 0 0] = [24 -84 60].
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 *)
CROSSREFS
Sequence in context: A143770 A062478 A005430 * A001574 A074445 A038154
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, May 02 2004
EXTENSIONS
More terms from Robert G. Wilson v, May 08 2004
STATUS
approved

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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)