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%I #35 Apr 02 2019 17:07:49
%S 1,3,4,7,10,17,24,41,58,99,140,239,338,577,816,1393,1970,3363,4756,
%T 8119,11482,19601,27720,47321,66922,114243,161564,275807,390050,
%U 665857,941664,1607521,2273378,3880899,5488420,9369319,13250218,22619537,31988856,54608393
%N Numerators of principal and intermediate convergents to 2^(1/2).
%C Sequence is essentially A082766 (by omitting two terms A082766(0) and A082766(2)). - _L. Edson Jeffery_, Apr 04 2011
%C a(n) = A119016(n+2) for n>=2. - _Georg Fischer_, Oct 07 2018
%D Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
%H Colin Barker, <a href="/A143607/b143607.txt">Table of n, a(n) for n = 1..1000</a>
%H Clark Kimberling, <a href="https://doi.org/10.1007/s000170050020">Best lower and upper approximates to irrational numbers</a>, Elemente der Mathematik, 52 (1997) 122-126.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,1).
%F From _Colin Barker_, Jul 28 2017: (Start)
%F G.f.: x*(1 + x)*(1 + 2*x + x^3) / (1 - 2*x^2 - x^4).
%F a(n) = 2*a(n-2) + a(n-4) for n>5.
%F (End)
%e The principal and intermediate convergents to 2^(1/2) begin with 1/1, 3/2 4/3, 7/5, 10/7, ...
%p 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
%t CoefficientList[Series[(1 + x)*(1 + 2*x + x^3) / (1 - 2*x^2 - x^4), {x, 0, 50}], x] (* or *)
%t LinearRecurrence[{0, 2, 0, 1}, {1, 3, 4, 7, 10}, 40] (* _Stefano Spezia_, Oct 08 2018; signature amended by _Georg Fischer_, Apr 02 2019 *)
%o (PARI) Vec(x*(1 + x)*(1 + 2*x + x^3) / (1 - 2*x^2 - x^4) + O(x^60)) \\ _Colin Barker_, Jul 28 2017
%o (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
%Y Cf. A002965 (denominators), A082766, A119016.
%K nonn,frac,easy
%O 1,2
%A _Clark Kimberling_, Aug 27 2008
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