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A368340
Take the solution to Pellian equation x^2 - 8*n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = x, or 1 if n is twice a positive square. A368339 gives values of y.
1
3, 1, 5, 17, 19, 7, 15, 1, 17, 9, 197, 49, 51, 127, 11, 577, 35, 1, 37, 721, 13, 199, 24335, 97, 99, 649, 485, 15, 19603, 31, 63, 1, 65, 33, 251, 17, 3699, 57799, 53, 161, 163, 55, 10405, 77617, 19, 1151, 2143295, 4801, 99, 1, 101, 5201, 32080051, 1351, 21, 127
OFFSET
1,1
LINKS
Carlos Rivera, Problem 88. Follow-up to problem 63, The Prime Puzzles & Problems Connection.
FORMULA
a(n) = A002350(8*n).
a(n) = sqrt(8*n*A368339(n)^2 + 1).
a(A000217(n)) = 2*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(1), 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(x))};
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved