cp's OEIS Frontend

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

%I A077057 #30 Aug 09 2017 23:05:54
%S A077057 1,2,5,3,3,27,7,37,4,4,171,22,9,14,1193,5,5,553,16,6173,11,45,143,849,
%T A077057 6,6,18339,94,1893,103,13,33,2353,115,12703,7,7,67115,701,73,59,
%U A077057 1891117,15,551427,23,49771,39,4105015,8,8,24673,41,75585293,25,9095891,989,17,386,6445,87,771,1385
%N A077057 Minimal positive solution a(n) of Diophantine equation a(n)^2 - a(n)*b(n) - G(n)*b(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077058(n).
%C A077057 This equation can also be written as (2*a(n) - b(n))^2 - D(n)*b(n)^2 = +4 or -4 with D(n) := A077425(n) = 1 + 4*G(n).
%C A077057 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 A077057 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 A077057 O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).
%H A077057 Wolfdieter Lang, <a href="/A077057/a077057.pdf">Pairs of smallest positive solutions for +1 and -1 if they exist</a>.
%F A077057 a(n) = (A078361(n) + A077058(n)) / 2. [_Max Alekseyev_, Feb 06 2010]
%t A077057 g[n_] := Ceiling[Sqrt[n]] + n - 1; r[n_] := Reduce[an > 0 && bn > 0 && (an ^2 - an*bn - g[n]*bn^2 == 1 || an^2 - an*bn - g[n]*bn^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, 62}] (* _Jean-François Alcover_, Oct 04 2012 *)
%Y A077057 Cf. A077427, A217470.
%K A077057 nonn
%O A077057 1,2
%A A077057 _Wolfdieter Lang_, Nov 29 2002
%E A077057 More terms from _Max Alekseyev_, Feb 06 2010
%E A077057 a(9), a(33), a(54) corrected (after notice by Jean-François Alcover); a(58) through a(62) added. - _Wolfdieter Lang_, Oct 04 2012