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).

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, 15, 298, 16, 2, 5, 352, 17, 1856, 1, 1, 9384, 97, 10, 8, 253970, 2, 72664, 3, 6440, 5, 521904, 1, 1, 3034, 5, 9148450, 3, 1084152, 117, 2, 45, 746, 10, 88, 157, 126890, 1, 1

Offset: 1

Author

Wolfdieter Lang, Nov 29 2002

Keywords

• nonn

Comments

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).

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.

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.

References

• O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Programs

Mathematica

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 *)

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

Extensions

More terms from Max Alekseyev, Feb 06 2010