A068717 -id:A068717 - cp's OEIS frontend

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

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

A097343 Triangle read by rows in which row n gives Legendre symbol (k,p) for 0

Original entry on oeis.org

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

Offset: 2

Robert G. Wilson v, Aug 02 2004

Row sums = 0. (p,k)==k^((p-1)/2) (mod p). For example, row n=4 of the triangle (for the 4th prime p = 7) reads: 1,1,-1,1,-1,-1,0 because 1^3==1, 2^3==1, 3^3==-1, 4^3==1, 5^3==-1, 6^3==-1, 7^3==0 (mod 7). - Geoffrey Critzer, Apr 18 2015

			1,-1,0 ; # A102283
1,-1,-1,1,0; # A080891
1,1,-1,1,-1,-1,0; # A175629
1,-1,1,1,1,-1,-1,-1,1,-1,0; # A011582

Reinhard Zumkeller, Rows n = 2..75 of triangle, flattened
Haskell for Math, Number Theory Fundamentals
Wikipedia, Legendre symbol

See A226520 for another version.

Cf. A068717.

a097343 n k = a097343_tabf !! (n-2) !! (k-1)
a097343_row n = a097343_tabf !! (n-2)
a097343_tabf =
   map (\p -> map (flip legendreSymbol p) [1..p]) $ tail a000040_list
legendreSymbol a p = if a' == 0 then 0 else twoSymbol * oddSymbol where
   a' = a `mod` p
   (s,q) = a' `splitWith` 2
   twoSymbol = if (p `mod` 8) `elem` [1,7] || even s then 1 else -1
   oddSymbol = if q == 1 then 1 else qrMultiplier * legendreSymbol p q
   qrMultiplier = if p `mod` 4 == 3 && q `mod` 4 == 3 then -1 else 1
   splitWith n p = spw 0 n where
      spw s t = if m > 0 then (s, t) else spw (s + 1) t'
                where (t', m) = divMod t p
-- See link.  Reinhard Zumkeller, Feb 02 2014

with(numtheory):
T:= n-> (p-> seq(jacobi(k, p), k=1..p))(ithprime(n)):
seq(T(n), n=2..15);  # Alois P. Heinz, Apr 19 2015

Flatten[ Table[ JacobiSymbol[ Range[ Prime[n]], Prime[n]], {n, 2, 8}]]

(p, p)=0, all others are +- 1.

A068716 a(n) = 1 if x^2 + 1 = n * y^2 has infinitely many solutions in integers (x,y), otherwise a(n) = 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Frank Ellermann, Feb 25 2002

nonn

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

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. A068717, A067280, A006702, A006703.

a(n) = 1 - (A067280(n) mod 2 ).

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 31 2003

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

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A097343 Triangle read by rows in which row n gives Legendre symbol (k,p) for 0

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A068716 a(n) = 1 if x^2 + 1 = n * y^2 has infinitely many solutions in integers (x,y), otherwise a(n) = 0.

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