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A091650
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Let M = the 4 X 4 matrix [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / -1 -1 3 2]. Set seed vector = [1 1 1 1] = first row, then take M*[1 1 1 1] = [1 1 1 3] then M * [1 1 1 3], etc. Sequence gives terms in rightmost column.
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0
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1, 3, 7, 21, 59, 171, 491, 1415, 4073, 11729, 33771, 97241, 279993, 806209, 2321385, 6684163, 19246279, 55417453, 159568195, 459458307, 1322957467, 3809304207, 10968454313, 31582405473, 90937912211, 261845282321, 753953441489, 2170922412257, 6250921954449
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OFFSET
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1,2
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COMMENTS
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a(n)/a(n-1) tends to a 9-Gon diagonal.
1. The other 3 columns are offsets of 1, 3, 7, 21, 59, ... starting with 1's. 2. The characteristic equation of the 4 X 4 matrix is x^4 - 2x^3 - 3x^4 + x + 1 (coefficients may be found in A066170) with roots 2.879385241..., -1, -.5320888862... and .65270364466...An alternative matrix giving the same eigenvalues (refer to A046854) relates to the 9-Gon: [1 1 1 1 / 1 1 1 0 / 1 1 0 0 / 1 0 0 0] since the eigenvalue 2.8793852...is the longest diagonal of the 9-Gon given edge = 1. Or, 2.879385... = 1/(2Cos k*Pi/9), k = 4.
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LINKS
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FORMULA
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G.f.: x*(1+x-2*x^2-x^3)/(1-2*x-3*x^2+x^3+x^4). [Colin Barker, Jan 31 2012]
a(1)=1, a(2)=3, a(3)=7, a(4)=21, a(n)=2*a(n-1)+3*a(n-2)-a(n-3)-a(n-4) [From Harvey P. Dale, Feb 17 2012]
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EXAMPLE
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a(5) = 59 since M*[1 1 1 1] then 4 iterates = [3 7 21 59]. a(5) = rightmost term.
a(10)/a(9) = 11729/4073 = 2.8796955...
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MATHEMATICA
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Rest[CoefficientList[Series[x (1+x-2x^2-x^3)/(1-2x-3x^2+x^3+x^4), {x, 0, 40}], x]] (* or *) LinearRecurrence[{2, 3, -1, -1}, {1, 3, 7, 21}, 40] (* Harvey P. Dale, Feb 17 2012 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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More terms from Harvey P. Dale, Feb 17 2012
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STATUS
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approved
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