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A077058 Minimal positive solution a(n) of Diophantine equation b(n)^2 - b(n)*a(n) - G(n)*a(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077057(n).

%I A077058 #16 Jun 10 2018 21:12:11
%S A077058 1,1,2,1,1,8,2,10,1,1,40,5,2,3,250,1,1,106,3,1138,2,8,25,146,1,1,2968,
%T A077058 15,298,16,2,5,352,17,1856,1,1,9384,97,10,8,253970,2,72664,3,6440,5,
%U A077058 521904,1,1,3034,5,9148450,3,1084152,117,2,45,746,10,88,157,126890,1,1
%N A077058 Minimal positive solution a(n) of Diophantine equation b(n)^2 - b(n)*a(n) - G(n)*a(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077057(n).
%C A077058 This equation can also be written as (2*b(n)-a(n))^2 - D(n)*a(n)^2 = +4 or -4 with D(n) := A077425(n)=1+4*G(n).
%C A077058 This is from Perron's table (see reference p. 108, for n = 1..28) which gives the minimal x,y values which solve the above mentioned Diophantine equations.
%C A077058 For Pell equation x^2 - D*y^2 = +4, see A077428 and A078355. For Pell equation x^2 - D*y^2 = -4, see A078356 and A078357.
%D A077058 O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).
%t A077058 g[n_] := Ceiling[ Sqrt[n] ] + n - 1; r[n_] := Reduce[an > 0 && (bn^2 - bn *an - g[n]*an^2 == 1 || bn^2 - bn *an - g[n]*an^2 == - 1), {an, bn}, Integers] /. C -> c; ab[n_] := DeleteCases[ Flatten[ Table[{an, bn} /. {ToRules[r[n]]} // Simplify, {c[1], 0, 1}] , 1] , an | bn]; a[n_] := a[n] = Min[ ab[n][[All, 1]] ]; Table[ Print[{n, a[n]}]; a[n], {n, 1, 65}] (* _Jean-François Alcover_, Oct 03 2012 *)
%o A077058 (PARI) forstep(D=1,1000,4, if(issquare(D),next); u=bnfinit(x^2-D).fu[1]; k=1; while( denominator(t=polcoeff(lift(u^k),1)*2)>1, k++); print1(abs(t),", "); ) \\ _Max Alekseyev_, Feb 06 2010
%K A077058 nonn
%O A077058 1,3
%A A077058 _Wolfdieter Lang_, Nov 29 2002
%E A077058 More terms from _Max Alekseyev_, Feb 06 2010