This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A153194 #17 Jan 08 2018 01:58:31
%S A153194 10,11,13,14,38,40,44,46,84,87,93,96,148,152,160,164,230,235,245,250,
%T A153194 330,336,348,354,448,455,469,476,584,592,608,616,738,747,765,774,910,
%U A153194 920,940,950,1100,1111,1133,1144,1308,1320,1344,1356
%N A153194 Numbers such that the numerator of floor(sqrt(n))/n, when reduced to its lowest terms, is equal to 3.
%C A153194 Previous name was: "For each number of the sequence, n, consider the fraction of squares from 1 to n inclusive. For numbers in this sequence, that fraction, when reduced to its lowest terms will always have 3 in the numerator.".
%C A153194 To obtain similar fractions as above with a numerator of 1, for example 1/5 are square, there are three possible numbers, namely 15, 20, 25. In general it is fairly easy to show that for 1/k of the numbers up to n (1 to n inclusive) to be square, n takes one of the three values, k(k-2), k(k-1), k^2. This sequence looks at obtaining fractions of the form 3/k. Another sequence (A153192) looks at the 2/k case.
%H A153194 Vincenzo Librandi, <a href="/A153194/b153194.txt">Table of n, a(n) for n = 1..1000</a>
%H A153194 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-2,0,0,-1,1).
%F A153194 G.f.: -x*(4*x^4+x^3+2*x^2+x+10) / ((x-1)^3*(x+1)^2*(x^2+1)^2). - _Colin Barker_, Mar 28 2014
%F A153194 G.f.: x * (10 + x + 2*x^2 + x^3 + 4*x^4) / ((1 - x) * (1 - x^4)^2). - _Michael Somos_, Mar 28 2014
%F A153194 a(n+4) = 2*a(n) - a(n-4) + 18 if n>0. - _Michael Somos_, Mar 28 2014
%e A153194 For 38, there are 6 squares below it and 6/38=3/19.
%e A153194 For 164 there are 12 squares below it and 12/164=3/41.
%t A153194 CoefficientList[Series[(10 + x + 2 x^2 + x^3 + 4 x^4)/((1 - x) (1 - x^4)^2), {x, 0, 60}], x] (* _Vincenzo Librandi_, Mar 29 2014 *)
%o A153194 (PARI) isok(n) = numerator(sqrtint(n)/n) == 3 \\ _Michel Marcus_, Aug 05 2013
%o A153194 (PARI) Vec(-x*(4*x^4+x^3+2*x^2+x+10)/((x-1)^3*(x+1)^2*(x^2+1)^2) + O(x^100)) \\ _Colin Barker_, Mar 28 2014
%Y A153194 Cf. A153192.
%K A153194 nonn,easy
%O A153194 1,1
%A A153194 _Anthony C Robin_, Dec 20 2008
%E A153194 Edited and more terms added by _Michel Marcus_, Aug 05 2013