login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A078356 Minimal positive solution z of Pell equation z^2 - A077426(n)*t^2 = -4. 7
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 (list; graph; refs; listen; history; text; internal format)
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
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
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 21 02:25 EST 2024. Contains 370219 sequences. (Running on oeis4.)