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%I #18 Mar 27 2016 22:38:21
%S 2,3,10,17,58,99,338,577,1970,3363,11482,19601,66922,114243,390050,
%T 665857,2273378,3880899,13250218,22619537,77227930,131836323,
%U 450117362,768398401,2623476242,4478554083,15290740090,26102926097,89120964298,152139002499
%N Numerators of the upper principal and intermediate convergents to 2^(1/2).
%C The upper principal and intermediate convergents to 2^(1/2), beginning with
%C 2/1, 3/2, 10/7, 17/12, 58/41, form a strictly decreasing sequence;
%C essentially, numerators=A143609 and denominators=A084068.
%D Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
%H Colin Barker, <a href="/A143609/b143609.txt">Table of n, a(n) for n = 1..1000</a>
%H Creighton Kenneth Dement, <a href="/A143608/a143608.txt">Comments on A143608 and A143609</a>
%H Clark Kimberling, <a href="http://dx.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 (0,6,0,-1).
%F a(n) = 6 * a(n-2) - a(n-4). a(2*n) = A001541(n) if n>0. a(2*n + 1) = 2 * A001653(n + 1).- _Michael Somos_, Sep 03 2013
%F G.f.: x * (2 + 3*x - 2*x^2 - x^3) / (1 - 6*x^2 + x^4). - _Michael Somos_, Sep 03 2013
%F a(n) = (2+sqrt(2)+(-1)^n*(-2+sqrt(2)))*((-1+sqrt(2))^n+(1+sqrt(2))^n)/(4*sqrt(2)). - _Colin Barker_, Mar 27 2016
%e 2*x + 3*x^2 + 10*x^3 + 17*x^4 + 58*x^5 + 99*x^6 + 338*x^7 + 577*x^8 + ...
%t Rest@ CoefficientList[Series[x (2 + 3 x - 2 x^2 - x^3)/(1 - 6 x^2 + x^4), {x, 0, 30}], x] (* _Michael De Vlieger_, Mar 27 2016 *)
%o (PARI) {a(n) = if( n<1, 0, polcoeff( x * (2 + 3*x - 2*x^2 - x^3) / (1 - 6*x^2 + x^4) + x * O(x^n), n))} /* _Michael Somos_, Sep 03 2013 */
%o (PARI) x='x+O('x^99); Vec(x*(2+3*x-2*x^2-x^3)/(1-6*x^2+x^4)) \\ _Altug Alkan_, Mar 27 2016
%Y Cf. A001541, A001653, A010914, A084068.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, Aug 27 2008
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