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a(n) is twice the coefficient of 1 in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 1).
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%I #47 Sep 08 2021 13:19:59

%S 2,4,1,10,16,2,6,20,4,3,30,8,8,34,340,4,5,394,48,10,10,52,16,5,22,

%T 3040,6,46,70,12,12,74,50,6,64,26,6964,20,7,48670,96,14,14,100,36,7,

%U 970,178,30,302,198,1060,8,39,126,16,16,130,97684,8,25,502,6960,34

%N a(n) is twice the coefficient of 1 in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 1).

%C From _Sean A. Irvine_, Jul 16 2021: (Start)

%C These values are computed by Algorithm 5.7.2 in Cohen.

%C Other methods of computation (see A346419) give different results, with the first difference at n=14.

%C (End)

%C a(n) is the smallest positive integer x satisfying the Pell equation x^2 - D*y^2 = +-4, where D = A000037(n). - _Jinyuan Wang_, Sep 08 2021

%D Henri Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, 1993.

%H Jinyuan Wang, <a href="/A048941/b048941.txt">Table of n, a(n) for n = 1..1000</a>

%H S. R. Finch, <a href="https://web.archive.org/web/20160511035657/http://www.people.fas.harvard.edu/~sfinch/csolve/clss.pdf">Class number theory</a>

%H Steven R. Finch, <a href="/A000924/a000924.pdf">Class number theory</a> [Cached copy, with permission of the author]

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a048/A048941.java">Java program</a> (github)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FundamentalUnit.html">Fundamental Unit</a>.

%o (PARI) a(n) = my(A, D=n+(1+sqrtint(4*n))\2, d=sqrtint(D), p, q, t, u1, u2, v1, v2); if(d%2==D%2, p=d, p=d-1); u1=-p; u2=2; v1=1; v2=0; q=2; while(v2==0 || q!=t, A=(p+d)\q; t=p; p=A*q-p; if(t==p && v2!=0, return((u2^2+D*v2^2)/q), t=A*u2+u1; u1=u2; u2=t; t=A*v2+v1; v1=v2; v2=t; t=q; q=(D-p^2)/q)); (u1*u2+D*v1*v2)/q; \\ _Jinyuan Wang_, Sep 08 2021

%Y Cf. A000037, A048942, A346419, A346420.

%K nonn

%O 1,1

%A _Eric W. Weisstein_

%E Name edited by _Michel Marcus_, Jun 26 2020

%E Entry revised by _Sean A. Irvine_, Jul 13 2021