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A145051 Numerator of the first convergent to sqrt(n) using the recursion x = (n/x + x)/2.

%I A145051 #43 Aug 17 2025 07:59:17
%S A145051 1,3,2,5,3,7,4,9,5,11,6,13,7,15,8,17,9,19,10,21,11,23,12,25,13,27,14,
%T A145051 29,15,31,16,33,17,35,18,37,19,39,20,41,21,43,22,45,23,47,24,49,25,51,
%U A145051 26,53,27,55,28,57,29,59,30,61,31,63,32,65,33,67,34,69,35,71,36,73,37,75
%N A145051 Numerator of the first convergent to sqrt(n) using the recursion x = (n/x + x)/2.
%C A145051 This is the same as A026741 without the first 2 terms in A026741. The link describes the experimental derivation of the generating function.
%C A145051 From _Jaroslav Krizek_, May 28 2010: (Start)
%C A145051 Numerators of arithmetic means of the first n positive integers for n >= 1.
%C A145051 See A040001 - denominators of arithmetic means of the first n positive integers.
%C A145051 a(n) = A026741(n+1) = A000217(n) * A040001(n) / n. (End)
%C A145051 Minimum number of line segments to draw into a circle to partition the circle into n+1 congruent circular sectors, i.e., minimum number of straight cuts required to cut a circular cake into n+1 equal slices. - _Felix Fröhlich_, Sep 01 2015
%C A145051 Continued fraction expansion of A386934. - _Kelvin Voskuijl_, Aug 15 2025
%H A145051 Kelvin Voskuijl, <a href="/A145051/b145051.txt">Table of n, a(n) for n = 1..10000</a>
%H A145051 Cino Hilliard, <a href="http://groups.google.com/group/roots-by-recursion/web/recursion-on-polynomial">Roots by Recursion</a> [Broken link]
%H A145051 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A145051 From _Paul Barry_, Nov 22 2009: (Start)
%F A145051 G.f.: x*(1 + 3*x - x^3)/(1 - x^2)^2.
%F A145051 a(n+1) = (n + 2)*(3 - (-1)^n)/4;
%F A145051 a(n+1) = Sum_{k=0..n, if(k=floor(n/2) or k=floor((n+1)/2),1,0)*(k+1)}. (End)
%F A145051 E.g.f.: ((x + 2)*cosh(x) + (2*x + 1)*sinh(x) - 2)/2. - _Stefano Spezia_, Apr 04 2024
%e A145051 n=1, x=1; x = (1/1+1)/2 = 1/1;
%e A145051 n=2, x=1; x = (2/1+1)/2 = 3/2;
%e A145051 n=3, x=1; x = (3/1+1)/2 = 2/1.
%e A145051 G.f.: x + 3*x^2 + 2*x^3 + 5*x^4 + 3*x^5 + 7*x^6 + 4*x^7 + 9*x^8 + 5*x^9 + ...
%t A145051 lst={};Do[a=n^2+n;b=n^2-n;c=a/b;AppendTo[lst,Denominator[c]],{n,2,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Oct 20 2009 *)
%o A145051 (PARI) g(n, p) = x=1;for(j=1,p,x=(n/x+x)/2; if(j==1, print1(numerator(x), ",")))
%o A145051 for(k=1,100,g(k,1))
%o A145051 (Magma) [(n+1)*(3 - (-1)^(n-1))/4: n in [1..100]]; // _Vincenzo Librandi_, Sep 02 2015
%Y A145051 Cf. A000217, A026741, A040001.
%K A145051 nonn,frac,easy
%O A145051 1,2
%A A145051 _Cino Hilliard_, Sep 30 2008