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A249581
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List of quadruples (r,s,t,u): the matrix M = [[9,24,16][3,10,8][1,4,4]] is raised to successive powers, then (r,s,t,u) are the square roots of M[3,1], M[3,3], M[1,1], M[1,3] respectively.
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2
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0, 1, 1, 0, 1, 2, 3, 4, 5, 8, 13, 20, 23, 36, 59, 92, 105, 164, 269, 420, 479, 748, 1227, 1916, 2185, 3412, 5597, 8740, 9967, 15564, 25531, 39868, 45465, 70996, 116461, 181860, 207391, 323852, 531243, 829564, 946025, 1477268, 2423293, 3784100
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OFFSET
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0,6
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COMMENTS
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The general form of these matrices is [[t^2,2tu,u^2][rt,st+ru,su][r^2,2rs,s^2]]. Different symmetries have different properties.
Iff |r * u - s * t| = 1 then terms to the left of a(0) are all integers.
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LINKS
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FORMULA
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a(4n) + a(4n + 1) = a(4n + 2).
a(4n + 1) + a(4n + 2) + a(4n + 3) - a(4n) = a(4n + 5)
4a(4n) = a(4n+3).
G.f.: x*(4*x^6-2*x^5-3*x^4+x^3+x+1) / (2*x^8-5*x^4+1). - Colin Barker, Nov 04 2014
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EXAMPLE
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M^0 = [[1,0,0][0,1,0][0,0,1]]. r = sqrt(M[3,1]) = a(0) = 0; s = sqrt(M[3,3]) = a(1) = 1; t = sqrt(M[1,1]) = a(2) = 1; u = sqrt(M[1,3]) = a(3) = 0.
M^1 = [[9,24,16][3,10,8][1,4,4]]. r = sqrt(M[3,1]) = a(4) = 1; s = sqrt(M[3,3]) = a(5) = 2; t = sqrt(M[1,1]) = a(6) = 3; u = sqrt(M[1,3]) = a(7) = 4.
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MATHEMATICA
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LinearRecurrence[{0, 0, 0, 5, 0, 0, 0, -2}, {0, 1, 1, 0, 1, 2, 3, 4}, 50] (* Harvey P. Dale, Aug 01 2016 *)
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PROG
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(PARI) concat(0, Vec(x*(4*x^6-2*x^5-3*x^4+x^3+x+1)/(2*x^8-5*x^4+1) + O(x^100))) \\ Colin Barker, Nov 04 2014
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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