OFFSET
1,2
COMMENTS
5x^2 - y^2 has discriminant 20, x^2 + xy - y^2 has discriminant 5. - N. J. A. Sloane, May 30 2014
Representable as x^2 + 3xy + y^2 with 0 <= x <= y. - Benoit Cloitre, Nov 16 2003
Numbers k such that x^2 - 3xy + y^2 + k = 0 has integer solutions. - Colin Barker, Feb 04 2014
Numbers k such that x^2 - 7xy + y^2 + 9k = 0 has integer solutions. - Colin Barker, Feb 10 2014
Also positive numbers of the form x^2 - 5y^2. - Jon E. Schoenfield, Jun 03 2022
REFERENCES
M. Baake, "Solution of coincidence problem ...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.
LINKS
Robert Israel and Vincenzo Librandi, Table of n, a(n) for n = 1..10000
M. Baake, Solution of the coincidence problem in dimensions d <= 4, arxiv:math/0605222 [math.MG], 2006.
M. Baake and R. V. Moody, Similarity submodules and semigroups in Quasicrystals and Discrete Geometry, ed. J. Patera, Fields Institute Monographs, vol. 10 AMS, Providence, RI (1998) pp. 1-13.
J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).
Norbert Hungerbühler and Maciej Smela, Geometric approach to the Diophantine equation x^2 + x*y - y^2 = m, hal-04835410, 2024. See p. 18.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
FORMULA
Consists exactly of numbers in which primes == 2 or 3 mod 5 occur with even exponents.
Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s)+Kronecker(m, p)*p^(-2s))^(-1) for m = 5.
MAPLE
select(t -> nops([isolve(5*x^2-y^2=t)])>0, [$1..1000]); # Robert Israel, Jun 12 2014
MATHEMATICA
ok[n_] := Resolve[Exists[{x, y}, Element[x|y, Integers], n == 5*x^2-y^2]]; Select[Range[236], ok]
(* or, for a large number of terms: *)
max = 60755 (* max=60755 yields 10000 terms *); A031363 = {}; xm = 1;
While[T = A031363; A031363 = Table[5*x^2 - y^2, {x, 1, xm}, {y, 0, Floor[ x*Sqrt[5]]}] // Flatten // Union // Select[#, # <= max&]&; A031363 != T, xm = 2*xm]; A031363 (* Jean-François Alcover, Mar 21 2011, updated Mar 17 2018 *)
PROG
(PARI) select(x -> x, direuler(p=2, 101, 1/(1-(kronecker(5, p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020, after hints by Colin Barker, Jun 18 2014, and Michel Marcus
(PARI) is(n)=#bnfisintnorm(bnfinit(z^2-z-1), n) \\ Ralf Stephan, Oct 18 2013
(PARI)
seq(M, k=3) = { \\ assume k >= 0
setintersect([1..M], setbinop((x, y)->x^2 + k*x*y + y^2, [0..1+sqrtint(M)]));
};
seq(236) \\ Gheorghe Coserea, Jul 29 2018
(Python)
from itertools import count, islice
from sympy import factorint
def A031363_gen(): # generator of terms
return filter(lambda n:all(not((1 < p % 5 < 4) and e & 1) for p, e in factorint(n).items()), count(1))
CROSSREFS
Numbers representable as x^2 + k*x*y + y^2 with 0 <= x <= y, for k=0..9: A001481(k=0), A003136(k=1), A000290(k=2), this sequence, A084916(k=4), A243172(k=5), A242663(k=6), A243174(k=7), A243188(k=8), A316621(k=9).
See A035187 for number of representations.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
See also the related sequence A263849 based on a theorem of Maass.
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Erich Friedman
b-file corrected and extended by Robert Israel, Jun 12 2014
STATUS
approved