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A249576
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List of triples (r,s,t): the matrix M = [[1,4,4][1,3,2][1,2,1]] is raised to successive powers, then (r,s,t) are the square roots of M[3,1], M[1,1], M[1,3] respectively.
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3
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0, 1, 0, 1, 1, 2, 2, 3, 4, 5, 7, 10, 12, 17, 24, 29, 41, 58, 70, 99, 140, 169, 239, 338, 408, 577, 816, 985, 1393, 1970, 2378, 3363, 4756, 5741, 8119, 11482, 13860, 19601, 27720, 33461, 47321, 66922, 80782, 114243, 161564, 195025, 275807, 390050, 470832, 665857, 941664
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OFFSET
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0,6
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COMMENTS
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Numbers to the left of a(0) are in A249577.
Some identities:
a(3n - 2) + a(3n - 1) = a(3n + 1).
a(3n) + a(3n + 1) = a(3(n + 1)).
a(3n - 2) + a(3n + 1) = a(3n + 2).
a(3n) + a(3n - 1) + a(3(n - 2)) = a(3n + 1).
a(3n - 1)a(3n) + a(3n + 2)a(3(n + 1)) = a(6n + 2).
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LINKS
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FORMULA
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a(n) = -2*a(n-3)+a(n-6); G.f.: -x*(2*x^4-x^3+x^2+1) / (x^6+2*x^3-1). - Colin Barker, Nov 02 2014
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EXAMPLE
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M^0 = the 3 X 3 identity matrix = [[1,0,0][0,1,0][0,0,1]]. M[3,1] = 0; M[1,1] = 1; M[1,3] = 0. So the first triple is r = a(0) = 0; s = a(1) = 1; t = a(2) = 0.
M^1 = [[1,4,4][1,3,2][1,2,1], so r = a(3) = 1; s = a(4) = 1; t = a(5) = 2.
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MATHEMATICA
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LinearRecurrence[{0, 0, 2, 0, 0, 1}, {0, 1, 0, 1, 1, 2}, 60] (* Harvey P. Dale, Dec 29 2021 *)
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PROG
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(PARI) concat(0, Vec(-x*(2*x^4-x^3+x^2+1)/(x^6+2*x^3-1) + O(x^100))) \\ Colin Barker, Nov 02 2014
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CROSSREFS
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a(3n) = the n-th term of A000129, the Pell numbers.
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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STATUS
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approved
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