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 A121234 #26 Oct 16 2024 20:33:11
%S A121234 0,2,4,55
%N A121234 Nonnegative integers k such that k*(k+1)*(k+2)+1 is a square.
%C A121234 The sequence is finite by Thue's theorem n(n + 1)(n + 2) + 1 = n^3 + 3*n^2 + 2*n + 1. The set of k values of integral solutions to the elliptic curve y^2 = n^3 + 3*n^2 + 2*n + 1 (see Magma program) is { -2, -1, 0, 2, 4, 55 }. So the sequence is complete. - _Mohamed Bouhamida_, Nov 29 2007
%e A121234 2 * 3 * 4 + 1 = 25 = 5^2, so 2 is in the sequence.
%e A121234 4 * 5 * 6 + 1 = 121 = 11^2, so 4 is in the sequence.
%e A121234 6 * 7 * 8 + 1 = 337, which is a prime number, so 6 is not in the sequence.
%t A121234 Select[Range[0,10^5], IntegerQ[Sqrt[#(#+1)(#+2)+1]]&] (* _James C. McMahon_, Oct 16 2024 *)
%o A121234 (Magma) P<n> := PolynomialRing(Integers()); {k: k in Sort([ p[1] : p in IntegralPoints(EllipticCurve(n^3 + 3*n^2 + 2*n + 1)) ])}; // _Mohamed Bouhamida_, Nov 29 2007
%o A121234 (PARI) isok(k) = issquare(k*(k+1)*(k+2)+1); \\ _Altug Alkan_, Dec 07 2015
%Y A121234 Cf. A258692.
%K A121234 fini,nonn,full
%O A121234 1,2
%A A121234 _Roger Cuculière_, Aug 21 2006