A368339 - OEIS

A368339

Take the solution to Pellian equation x^2 - 8*n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is twice a positive square. A368340 gives values of x.

1, 0, 1, 3, 3, 1, 2, 0, 2, 1, 21, 5, 5, 12, 1, 51, 3, 0, 3, 57, 1, 15, 1794, 7, 7, 45, 33, 1, 1287, 2, 4, 0, 4, 2, 15, 1, 215, 3315, 3, 9, 9, 3, 561, 4137, 1, 60, 110532, 245, 5, 0, 5, 255, 1557945, 65, 1, 6, 48, 455, 14127, 11, 11, 207480, 20, 29427, 285, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..66.

Carlos Rivera, Problem 88. Follow-up to problem 63, The Prime Puzzles & Problems Connection.

FORMULA

a(n) = A002349(8*n).

a(n) = sqrt((A368340(n)^2 - 1)/(8*n)).

a(A000217(n)) = 1, n >= 1.

EXAMPLE

For n = 1, 2, 3, 4, 5 solutions are (x,y) = (3, 1), (1, 0), (5, 1), (17, 3), (19, 3).

PROG

(PARI) pellsolve(n)={if(issquare(n/2), return(0), q=bnfinit('x^2-8*n, 1); i=-1; until(y&&x==floor(x)&&y==floor(y)&&x^2-8*n*y^2==1, f=lift(q.fu[1]^i); x=abs(polcoeff(f, 0)); y=abs(polcoeff(f, 1)); i++); return(y))};

CROSSREFS

Cf. A000217, A002349, A268856, A368340.

Sequence in context: A100940 A344390 A063421 * A244328 A073067 A003637

Adjacent sequences: A368336 A368337 A368338 * A368340 A368341 A368342

KEYWORD

nonn,easy

AUTHOR

Arkadiusz Wesolowski, Dec 21 2023

STATUS

approved