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A339326 Denominators of s in rational solutions of s^4 + s^3 + s^2 + s + 1 = y^2 with |s| <= 1.

%I A339326 #17 Nov 15 2021 09:32:22
%S A339326 1,1,3,11,123,808,43993,1404304,113095273,16258517264,5907678749271,
%T A339326 1749162037068984,2230703155726839733,2430407134728632414424,
%U A339326 9811722627654286580946253,28104484948123389151863529007,447820184835469405718954028342863,5093605667758828993168776807306887631
%N A339326 Denominators of s in rational solutions of s^4 + s^3 + s^2 + s + 1 = y^2 with |s| <= 1.
%C A339326 If abs(A339325(n)) = 1 or a(n) = 1 then n <= 3, i.e., the only integer solutions to s^4 + s^3 + s^2 + s + 1 = y^2 are (s, y) = (-1, +-1), (0, +-1), (3, +-11). This may easily be shown by bounding the LHS between two consecutive perfect squares.
%H A339326 Jeremy Tan, <a href="/A339326/b339326.txt">Table of n, a(n) for n = 1..50</a>
%H A339326 Jeremy Tan, <a href="https://math.stackexchange.com/q/3604438">Rigid pentagons and rational solutions of s^4+s^3+s^2+s+1=y^2</a>, Mathematics Stack Exchange, Apr 1 2020.
%H A339326 Gerard 't Hooft, <a href="https://webspace.science.uu.nl/~hooft101/lectures/meccano.pdf">Meccano Math I</a>
%e A339326 The values of s in solutions (s, y) with |s| <= 1 begin -1, 0, 1/3, -8/11, -35/123, 627/808, -20965/43993, ...
%t A339326 a[1] = 1; a[2] = 1; a[n_] := Module[{x = 4, y = 2, s, xr}, Do[s = (y-1) / (x-1); xr = s^2 - x + 4; {x, y} = {xr, s(x-xr) - y}, n-2]; s = (2y-x) / (4x-5); Denominator[MinimalBy[{s, 1/s}, Abs][[1]]]]; Table[a[k], {k, 20}] (* _Jeremy Tan_, Nov 15 2021 *)
%o A339326 (PARI)
%o A339326 a(n) = {
%o A339326     [u,v] = ellmul(ellinit([0,-5,0,5,0]), [1,1], n);
%o A339326     s = (2*v-u) / (4*u-5);
%o A339326     if(abs(s)>1, s=1/s);
%o A339326     denominator(s)
%o A339326 }
%Y A339326 Cf. A339325 (numerators).
%K A339326 nonn,frac
%O A339326 1,3
%A A339326 _Jeremy Tan_, Nov 30 2020