A031363 Positive numbers of the form x^2 + xy - y^2; or, of the form 5x^2 - y^2.

1, 4, 5, 9, 11, 16, 19, 20, 25, 29, 31, 36, 41, 44, 45, 49, 55, 59, 61, 64, 71, 76, 79, 80, 81, 89, 95, 99, 100, 101, 109, 116, 121, 124, 125, 131, 139, 144, 145, 149, 151, 155, 164, 169, 171, 176, 179, 180, 181, 191, 196, 199, 205, 209, 211, 220, 225, 229, 236

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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)

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.

Primes in this sequence: A038872, also A141158.

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.

Programs

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 *)

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))
A031363_list = list(islice(A031363_gen(),30)) # Chai Wah Wu, Jun 28 2022

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.

Extensions

More terms from Erich Friedman

b-file corrected and extended by Robert Israel, Jun 12 2014