|
| |
|
|
A074074
|
|
The numbers D in the set {D :=(2n+1)^2-4m^2, 1<=m<=n} that generate the smallest solution x to x^2 - D*y^2 = 1.
|
|
3
|
|
|
|
5, 21, 33, 17, 105, 105, 189, 33, 105, 405, 333, 141, 473, 57, 817, 189, 325, 885, 77, 1425, 1173, 1925, 1425, 2301, 101, 105, 1425, 333, 777, 1785, 2525, 381, 1785, 405, 141, 3393, 2261, 3813, 6045, 2717, 4389, 3129, 2093, 6765, 885, 5513, 189, 6045, 197, 8085
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
Given a discriminant D, the Pell equation x^2-D*y^2=1 has a minimum solution x
as tabulated in A033313. We start with a set of candidates D of the form
(2*n+1)^2-(2*m)^2, obviously all odd, where m runs through the integers from 1 to n.
Whichever D out of this set generates the smallest x in A033313, defines a(n)=D.
|
|
|
LINKS
|
Ray Chandler, Table of n, a(n) for n = 1..499
|
|
|
EXAMPLE
|
For n=4, the candidates are D=77 (m=1, index 69 in A000037), D=65 (m=2, index 57 in A000037),
D=45 (m=3, index 39 in A000037) and D=17 (m=4, index 13 in A000037),
which produce x = 351, x=129, x=161 and x=33 in that order (apply the offset in A033313
while converting indices from A000037 to find the x). Because 33 is the smallest
of these four x, we select the associated D=17 as a(n=4).
|
|
|
MAPLE
|
A033313 := proc(Dcap) local c, i, fr, nu, de ; if issqr(Dcap) then -1; else c := numtheory[cfrac](sqrt(Dcap)) ; for i from 1 do try fr := numtheory[nthconver](c, i) ; nu := numer(fr) ; de := denom(fr) ; if nu^2-Dcap*de^2=1 then RETURN(nu) ; fi; catch: RETURN(-1) ; end try; od: fi: end:
A074074 := proc(n) local Dmin, xmin, Dcap ; Dmin := -1 ; xmin := -1; for m from 1 to n do Dcap := (2*n+1+2*m)*(2*n+1-2*m) ; x := A033313(Dcap) ; if xmin = -1 or (x >0 and x<xmin ) then Dmin := Dcap ; xmin := x ; fi; od: Dmin ; end:
seq(A074074(n), n=1..50) ; # R. J. Mathar, Sep 21 2009
|
|
|
CROSSREFS
|
Cf. A074075 (associated m values), A074076.
Sequence in context: A305645 A316921 A316423 * A006309 A273577 A317039
Adjacent sequences: A074071 A074072 A074073 * A074075 A074076 A074077
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Lekraj Beedassy, Aug 28 2002
|
|
|
EXTENSIONS
|
Definition clarified, sequence extended beyond a(7) - R. J. Mathar, Sep 21 2009
|
|
|
STATUS
|
approved
|
| |
|
|
