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A143607
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Numerators of principal and intermediate convergents to 2^(1/2).
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6
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1, 3, 4, 7, 10, 17, 24, 41, 58, 99, 140, 239, 338, 577, 816, 1393, 1970, 3363, 4756, 8119, 11482, 19601, 27720, 47321, 66922, 114243, 161564, 275807, 390050, 665857, 941664, 1607521, 2273378, 3880899, 5488420, 9369319, 13250218, 22619537, 31988856, 54608393
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OFFSET
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1,2
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COMMENTS
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REFERENCES
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Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
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LINKS
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FORMULA
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G.f.: x*(1 + x)*(1 + 2*x + x^3) / (1 - 2*x^2 - x^4).
a(n) = 2*a(n-2) + a(n-4) for n>5.
(End)
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EXAMPLE
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The principal and intermediate convergents to 2^(1/2) begin with 1/1, 3/2 4/3, 7/5, 10/7, ...
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MAPLE
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seq(coeff(series(x*(1+x)*(1+2*x+x^3)/(1-2*x^2-x^4), x, n+1), x, n), n = 1 .. 40); # Muniru A Asiru, Oct 07 2018
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MATHEMATICA
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CoefficientList[Series[(1 + x)*(1 + 2*x + x^3) / (1 - 2*x^2 - x^4), {x, 0, 50}], x] (* or *)
LinearRecurrence[{0, 2, 0, 1}, {1, 3, 4, 7, 10}, 40] (* Stefano Spezia, Oct 08 2018; signature amended by Georg Fischer, Apr 02 2019 *)
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PROG
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(PARI) Vec(x*(1 + x)*(1 + 2*x + x^3) / (1 - 2*x^2 - x^4) + O(x^60)) \\ Colin Barker, Jul 28 2017
(GAP) a:=[1, 3, 4, 7, 10];; for n in [6..40] do a[n]:=2*a[n-2]+a[n-4]; od; a; # Muniru A Asiru, Oct 07 2018
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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