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A368339 Take the solution to Pellian equation x^2 - 8*n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is twice a positive square. A368340 gives values of x.

%I A368339 #20 Jan 25 2024 07:27:13
%S A368339 1,0,1,3,3,1,2,0,2,1,21,5,5,12,1,51,3,0,3,57,1,15,1794,7,7,45,33,1,
%T A368339 1287,2,4,0,4,2,15,1,215,3315,3,9,9,3,561,4137,1,60,110532,245,5,0,5,
%U A368339 255,1557945,65,1,6,48,455,14127,11,11,207480,20,29427,285,1
%N A368339 Take the solution to Pellian equation x^2 - 8*n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is twice a positive square. A368340 gives values of x.
%H A368339 Carlos Rivera, <a href="https://primepuzzles.net/problems/prob_088.htm">Problem 88. Follow-up to problem 63</a>, The Prime Puzzles & Problems Connection.
%F A368339 a(n) = A002349(8*n).
%F A368339 a(n) = sqrt((A368340(n)^2 - 1)/(8*n)).
%F A368339 a(A000217(n)) = 1, n >= 1.
%e A368339 For n = 1, 2, 3, 4, 5 solutions are (x,y) = (3, 1), (1, 0), (5, 1), (17, 3), (19, 3).
%o A368339 (PARI) pellsolve(n)={if(issquare(n/2), return(0), q=bnfinit('x^2-8*n, 1); i=-1; until(y&&x==floor(x)&&y==floor(y)&&x^2-8*n*y^2==1, f=lift(q.fu[1]^i); x=abs(polcoeff(f, 0)); y=abs(polcoeff(f, 1)); i++); return(y))};
%Y A368339 Cf. A000217, A002349, A268856, A368340.
%K A368339 nonn,easy
%O A368339 1,4
%A A368339 _Arkadiusz Wesolowski_, Dec 21 2023