A078356 Minimal positive solution z of Pell equation z^2 - A077426(n)*t^2 = -4.

1, 3, 8, 5, 12, 64, 7, 39, 16, 2136, 9, 1000, 11208, 20, 261, 1552, 11, 3488, 24, 61, 213, 13, 1305, 136, 3528264, 28, 15, 46312, 142022136, 32, 12144, 164, 2613, 2127064, 17, 253724736, 89, 36, 2031654672, 18420, 142528, 19, 10236, 2564, 3447, 40, 223843593936

Offset

1,2

Comments

The corresponding values of t are given in A078357.

Computed from Perron's table (see reference p. 108) which gives the minimal x,y values for the Diophantine equation x^2 - x*y - ((D(m)-1)/4)*y^2 = +1 and -1 for respectively D(m)=A077425(m) and D(m)=A077426(m) (this second case excludes in Perron's table the D values with a 'Teilnenner' in brackets).

The conversion from the x,y values of Perron's table to the minimal a=a(n) and b=b(n) solutions of a^2 - D(n)*b^2 =-4 see a comment in A077428. Here only D values with no 'Teilnenner' in brackets are of interest and a(n)=2*x(n)-y(n) and b(n)=y(n). E.g. D=41, with 'Teilnenner von (sqrt(D)+1)/2' in the notation, explained in an example of A077427, 3,1,2 (period length k=5) and (x,y)=(37,10) which translates to the minimal solution (a,b)=(64,10).

Generic D(n) values are those from A078370(k)=(4*k(k+1)+5), k>=0, which are 5 (mod 8). For such D values the minimal solution is (a,b)=(2*k+1,1) (e.g. D(7)= A077426(7) = 53 = A078370(3) with a(7)= 2*3+1=7 and b(7)=A078357(7)=1).

The general solution of Pell a^2-D(n)*b^2 = -4 with generic D(n)=A078370(k), k>=0, is a(n,m)= (2*k+1)*S(2*m,sqrt(D(n))) and b(n,m)= T(2*m+1,sqrt(D(n))/2)/(sqrt(D(n))/2), m>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 resp. A049310.

For non-generic D(n) (not from A078370) the general solution of a^2-D(n)*b^2 = -4 is a(n,m)=a(n)*S(2*m,sqrt(a(n)^2+4)) and b(n,m)= b(n)*T(2*m+1,sqrt(a(n)^2+4)/2)/(sqrt(a(n)^2+4)/2), m>=0, with Chebyshev's polynomials and in this case b(n)>1.

References

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Index entries for sequences related to Chebyshev polynomials.

Example

41=D(6)=A077426(6) (also A077425(8)), hence a(6)=64 and b(6)=A078357(6)=10 satisfies 64^2 - 41*10^2 = -4.

Mathematica

$MaxExtraPrecision = 100; A077426 = Select[Range[ 500], ! IntegerQ[Sqrt[#]] && OddQ[ Length[ ContinuedFraction[(Sqrt[#] + 1)/2] // Last]] &]; a[n_] := {z, t} /. {ToRules[ Reduce[z > 0 && t > 0 && z^2 - A077426[[n]]*t^2 == -4, {z, t}, Integers] /. C[1] -> 0]} // Sort // First // First; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jun 21 2013 *)

Crossrefs

Sequence in context: A347942 A058055 A229598 * A050093 A120072 A166492

Adjacent sequences: A078353 A078354 A078355 * A078357 A078358 A078359

Keyword

nonn

Author

Wolfdieter Lang, Nov 29 2002

Extensions

More terms from R. J. Mathar, Sep 24 2009

Edited by Max Alekseyev, Mar 03 2010

Status

approved