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 A135509 #39 Feb 21 2022 02:17:59
%S A135509 0,1,25,225,1156,4225,12321,30625,67600,136161,255025,450241,756900,
%T A135509 1221025,1901641,2873025,4227136,6076225,8555625,11826721,16080100,
%U A135509 21538881,28462225,37149025,47941776,61230625,77457601,97121025
%N A135509 Nonnegative integers c such that there are nonnegative integers a and b that satisfy a^(1/2) + b^(1/2) = c^(1/2) and a^2 + b = c.
%C A135509 Define FLTR as Fermat's Last Theorem with rational exponents. Consider x + y = x + y. Then (x^m)^(1/m) + (y^m)^(1/m) = ((x+y)^m)^(1/m) for integer m >= 1.
%C A135509 For m = 2, we have (x^2)^(1/2) + (y^2)^(1/2) = ((x+y)^2)^(1/2). Thus, a = x^2, b = y^2 and c = (x+y)^2. Then a^2 + b = c => x^4 + y^2 = (x+y)^2 => x^4 + y^2 = x^2 + 2*x*y + y^2 => y = (x^3-x)/2. It follows that c = (x+y)^2 = (x^3 + x)^2/4 is the generating function for this sequence for x = 0, 1, 2, 3, ...
%C A135509 For m = 2, there are infinitely many nonnegative integer solutions for the FLTR proposition. The same holds for m = 3, i.e., there are also infinitely many nonnegative integer solutions to a^(1/3) + b^(1/3) = c^(1/3). E.g., 8^(1/3) + 27^(1/3) = 125^(1/3). Moreover, there are infinitely many solutions to FLTR for a general positive integer m.
%C A135509 However, in conjunction with a^2 + b = c, I could not find any nontrivial solutions when m >= 3. Perhaps there is another formula that will yield solutions. [Edited by _Petros Hadjicostas_, Dec 21 2019]
%H A135509 G. C. Greubel, <a href="/A135509/b135509.txt">Table of n, a(n) for n = 0..1000</a>
%H A135509 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A135509 From _Colin Barker_, May 02 2012: (Start)
%F A135509 a(n) = (n + n^3)^2/4 = A006003(n)^2.
%F A135509 G.f.: x*(1 + x)*(1 + 4*x + x^2)*(1 + 13*x + x^2)/(1 - x)^7. (End)
%F A135509 E.g.f.: (1/4)*x*(4 + 46*x + 102*x^2 + 67*x^3 + 15*x^4 + x^5)*exp(x). - _G. C. Greubel_, Oct 16 2016
%e A135509 For a = 9, b = 144, and c = 225, we obtain 9^(1/2) + 144^(1/2) = 225^(1/2) and 9^2 + 144 = 225. Thus, c = 225 is an entry in this sequence.
%t A135509 Table[(n + n^3)^2/4, {n, 0, 25}] (* _G. C. Greubel_, Oct 16 2016 *)
%o A135509 (PARI) flt2(n) = {local(a, b); for(a=0, n, b = (a^3+a)/2; print1(b^2", "))} /* edited by _Petros Hadjicostas_, Dec 21 2019 */
%Y A135509 Cf. A006003, A135503.
%K A135509 nonn,easy
%O A135509 0,3
%A A135509 _Cino Hilliard_, Feb 09 2008