A068716 -id:A068716 - cp's OEIS frontend

A068717 a(n) = -1 if A067280(n) == 0 (mod 2), otherwise a(n) = A049240(n).

0, -1, 1, 0, -1, 1, 1, 1, 0, -1, 1, 1, -1, 1, 1, 0, -1, 1, 1, 1, 1, 1, 1, 1, 0, -1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 0, -1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 0, -1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 0, -1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 0, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, 1

Offset: 1

Frank Ellermann, Feb 25 2002

Previous name was: x*x - n*y*y = +-1 has infinitely many solutions in integers (x,y).

			a(2)= -1: x*x - 2*y*y = -1 is soluble, e.g., 7*7 - 2*5*5 = -1.

H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th ed., 1999, table 1.

John Robertson, Solving the generalized Pell equation x^2-dy^2=N.

Cf. A068716, A068718, A067280, A049240, A006702, A006703.

from math import isqrt
from sympy import continued_fraction_periodic
def A068717(n): return 0 if (a:=isqrt(n)**2==n) else (-1 if len(continued_fraction_periodic(0,1,n)[1]) & 1 else 1-int(a)) # Chai Wah Wu, Jun 14 2022

a(n) = -1 if A067280(n) == 0 (mod 2), otherwise a(n) = A049240(n).

New name from formula by Joerg Arndt, Aug 29 2020

A068718 Boustrophedon transform of A068717 with A068717(1) = -1 instead of 0.

Original entry on oeis.org

1, 2, 2, 2, 8, 25, 90, 404, 2055, 11792, 75053, 525622, 4015361, 33231679, 296182315, 2828335731, 28809181418, 311788442591, 3572832236720, 43216177809190, 550245463265240, 7356239983352887, 103028812072639378

Offset: 0

Frank Ellermann, Feb 26 2002

Signs omitted, all terms < 0. A068717(1) = -1 means soluble (1*1 -1*0*0 = +1 resp. 0*0 -1*1*1 = -1).

Cf. A068716, A068717, A049240, A046951.

from itertools import accumulate, count, islice
from math import isqrt
from sympy import continued_fraction_periodic
def A068718_gen(): # generator of terms
    yield 1
    blist = (1,)
    for n in count(2):
        yield (blist := tuple(accumulate(reversed(blist),initial=0 if (a:=isqrt(n)**2==n) else (1 if len(continued_fraction_periodic(0,1,n)[1]) & 1 else int(a)-1))))[-1]
A068718_list = list(islice(A068718_gen(),40)) # Chai Wah Wu, Jun 14 2022

cp's OEIS Frontend

A068717 a(n) = -1 if A067280(n) == 0 (mod 2), otherwise a(n) = A049240(n).

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