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%I #20 Sep 21 2015 17:49:37
%S 1,5,6,8,11,14,17,20,24,28,32,36,40,44,49,54,59,64,69,74,79,85,91,97,
%T 103,109,115,121,127,134,141,148,155,162,169,176,183,190,198,206,214,
%U 222,230,238,246,254,262,270,278,286,295,304,313,322,331,340,349,358,367,376,385,394,404,414,424,434,444,454,464,474,484,494,504,514,524,535,546,557,568,579,590,601,612,623,634,645,656,667,678,689,701,713,725,737,749,761,773,785,797,809
%N a(n) is the largest term in the continued fraction for a(n-1) + n^2/a(n-1), where a(1)=1.
%C There is an asymptotic formula for this sequence (see formula line).
%H Benoit Cloitre, <a href="/A262161/a262161.png">Plot of (a(n)-n*(sqrt(2*n/3)-1/5))/sqrt(n) for 10000<n<20000</a>
%F a(n) = n*(sqrt(2*n/3)-1/5) + O(sqrt(n)) and this is sharp (see link).
%e We have a(7)=17 and the continued fraction for a(7) + 8^2/a(7) is [20, 1, 3, 4], hence a(8)=20.
%t a = {1}; Do[AppendTo[a, Max@ ContinuedFraction[a[[n - 1]] + (n^2)/a[[n - 1]]]], {n, 2, 100}]; a (* _Michael De Vlieger_, Sep 14 2015 *)
%o (PARI) a=1;for(n=2,100,a=vecmax(contfrac(a+n^2/a));print1(a,","))
%K nonn
%O 1,2
%A _Benoit Cloitre_, Sep 13 2015