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

%I A368340 #19 Jan 25 2024 07:28:03
%S A368340 3,1,5,17,19,7,15,1,17,9,197,49,51,127,11,577,35,1,37,721,13,199,
%T A368340 24335,97,99,649,485,15,19603,31,63,1,65,33,251,17,3699,57799,53,161,
%U A368340 163,55,10405,77617,19,1151,2143295,4801,99,1,101,5201,32080051,1351,21,127
%N A368340 Take the solution to Pellian equation x^2 - 8*n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = x, or 1 if n is twice a positive square. A368339 gives values of y.
%H A368340 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 A368340 a(n) = A002350(8*n).
%F A368340 a(n) = sqrt(8*n*A368339(n)^2 + 1).
%F A368340 a(A000217(n)) = 2*n + 1, n >= 1.
%e A368340 For n = 1, 2, 3, 4, 5 solutions are (x,y) = (3, 1), (1, 0), (5, 1), (17, 3), (19, 3).
%o A368340 (PARI) pellsolve(n)={if(issquare(n/2), return(1), 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(x))};
%Y A368340 Cf. A000217, A002350, A069129, A368339.
%K A368340 nonn,easy
%O A368340 1,1
%A A368340 _Arkadiusz Wesolowski_, Dec 21 2023