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 A050271 #53 Aug 10 2025 11:52:33
%S A050271 1,2,3,4,7,8,9,14,15,16,23,24,25,34,35,36,47,48,49,62,63,64,79,80,81,
%T A050271 98,99,100,119,120,121,142,143,144,167,168,169,194,195,196,223,224,
%U A050271 225,254,255,256,287,288,289,322,323,324,359,360,361,398,399,400
%N A050271 Numbers k such that k = floor(sqrt(k)*ceiling(sqrt(k))).
%C A050271 Is a(n) asymptotic to C*n^(3/2) where 1/2 < C < 1?
%C A050271 Consists exactly of numbers of the forms j^2 - 2, j^2 - 1, and j^2. As such, is asymptotic to (1/9)*n^2. - _Ivan Neretin_, Feb 08 2017
%H A050271 Ivan Neretin, <a href="/A050271/b050271.txt">Table of n, a(n) for n = 1..10000</a>
%H A050271 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).
%F A050271 a(n) = floor((n + 4)/3)^2 + ((n + 1) mod 3) - 2. - _Ivan Neretin_, Feb 08 2017
%F A050271 From _Colin Barker_, Feb 09 2017: (Start)
%F A050271 a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n > 7.
%F A050271 G.f.: x*(1 + x + x^2 - x^3 + x^4 - x^5) / ((1 - x)^3*(1 + x + x^2)^2).
%F A050271 (End)
%F A050271 From _Amiram Eldar_, Sep 14 2022: (Start)
%F A050271 Sum_{n>=1} 1/a(n) = 2 + Pi^2/6 - cot(sqrt(2)*Pi)*Pi/(2*sqrt(2)).
%F A050271 Sum_{n>=1} (-1)^(n+1)/a(n) = 1 + Pi^2/12 + cosec(sqrt(2)*Pi)*Pi/(2*sqrt(2)). (End)
%F A050271 a(n) = A087278(n+1) - 1 if n > 0. - _Lorenzo Sauras Altuzarra_, Jan 31 2023
%p A050271 a:=n->floor((n+4)/3)^2+irem(n+1,3)-2:
%p A050271 seq(a(n),n=1..58); # _Lorenzo Sauras Altuzarra_, Jan 31 2023
%t A050271 Select[Range@400, Floor[(r = Sqrt@#)*Ceiling@r] == # &] (* _Ivan Neretin_, Feb 08 2017 *)
%t A050271 LinearRecurrence[{1,0,2,-2,0,-1,1},{1,2,3,4,7,8,9},60] (* _Harvey P. Dale_, Aug 10 2025 *)
%o A050271 (PARI) isok(n) = floor(sqrt(n)*ceil(sqrt(n))) == n; \\ _Michel Marcus_, Nov 22 2013
%o A050271 (PARI) Vec(x*(1 + x + x^2 - x^3 + x^4 - x^5) / ((1 - x)^3*(1 + x + x^2)^2) + O(x^100)) \\ _Colin Barker_, Feb 09 2017
%o A050271 (Python)
%o A050271 def A050271(n):
%o A050271 a, b = divmod(n+4,3)
%o A050271 return a**2+b-2 # _Chai Wah Wu_, Aug 02 2022
%Y A050271 Cf. A000290, A005563, A008865, A087278.
%K A050271 nonn,easy
%O A050271 1,2
%A A050271 _Benoit Cloitre_, May 10 2003
%E A050271 Data corrected by _Michel Marcus_ and _Benoit Cloitre_, Nov 22 2013