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 A153192 #24 Sep 23 2022 15:12:45
%S A153192 5,7,18,22,39,45,68,76,105,115,150,162,203,217,264,280,333,351,410,
%T A153192 430,495,517,588,612,689,715,798,826,915,945,1040,1072,1173,1207,1314,
%U A153192 1350,1463,1501,1620,1660,1785,1827,1958
%N A153192 Numbers such that the numerator of floor(sqrt(n))/n, when reduced to its lowest terms, is equal to 2.
%C A153192 Previous name was: "For each term in this sequence, n say, consider the fraction of square numbers between 1 & n, inclusive, when reduced to its lowest terms. This fraction will always have a numerator of 2 for numbers in this sequence".
%C A153192 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 2/k. Another sequence (A153194) looks at the 3/k case.
%C A153192 Alternately, numbers of the form 4n^2+n or 4n^2+3n. - _Charles R Greathouse IV_, Aug 05 2013
%H A153192 Charles R Greathouse IV, <a href="/A153192/b153192.txt">Table of n, a(n) for n = 1..10000</a>
%H A153192 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A153192 From _Colin Barker_, Mar 28 2014: (Start)
%F A153192 a(n) = (2*n+3)*(2*n-(-1)^n+1)/4.
%F A153192 a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
%F A153192 G.f.: -x*(x^2+2*x+5) / ((x-1)^3*(x+1)^2). (End).
%e A153192 22 has 4 square numbers below it and 4/22=2/11.
%e A153192 76 has 8 square numbers below it and 8/76=2/19.
%t A153192 LinearRecurrence[{1,2,-2,-1,1},{5,7,18,22,39},50] (* _Harvey P. Dale_, Sep 23 2022 *)
%o A153192 (PARI) isok(n) = numerator(sqrtint(n)/n) == 2 \\ _Michel Marcus_, Aug 05 2013
%o A153192 (PARI) for(n=1,9,print1(4*n^2+n", "4*n^2+3*n", ")) \\ _Charles R Greathouse IV_, Aug 05 2013
%Y A153192 Cf. A153194.
%K A153192 nonn,easy
%O A153192 1,1
%A A153192 _Anthony C Robin_, Dec 20 2008
%E A153192 Definition simplified and more terms added by _Michel Marcus_, Aug 05 2013