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%I #18 May 12 2018 17:33:39
%S 1,2,3,5,7,12,19,26,45,71,97,168,265,362,627,989,1351,2340,3691,5042,
%T 8733,13775,18817,32592,51409,70226,121635,191861,262087,453948,
%U 716035,978122,1694157,2672279,3650401,6322680,9973081,13623482,23596563,37220045
%N Numerators of principal and intermediate convergents to 3^(1/2).
%D Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
%H Colin Barker, <a href="/A143642/b143642.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_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,4,0,0,-1).
%F From _Colin Barker_, Jul 28 2017: (Start)
%F G.f.: x*(1 + x)*(1 + x + 2*x^2 - x^3) / (1 - 4*x^3 + x^6).
%F a(n) = 4*a(n-3) - a(n-6) for n>6.
%F (End)
%e The first few principal and intermediate convergents to 3^(1/2) are 1/1, 2/1, 3/2, 5/3, 7/4, 12/7, ...
%t LinearRecurrence[{0,0,4,0,0,-1},{1,2,3,5,7,12},40] (* _Harvey P. Dale_, May 12 2018 *)
%o (PARI) Vec(x*(1 + x)*(1 + x + 2*x^2 - x^3) / (1 - 4*x^3 + x^6) + O(x^60)) \\ _Colin Barker_, Jul 28 2017
%Y Cf. A140827 (denominators).
%K nonn,frac,easy
%O 1,2
%A _Clark Kimberling_, Aug 27 2008
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