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

1,2

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).

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

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.

Sequence in context: A117870 A162698 A166562 * A118142 A193584 A155149

Adjacent sequences: A031360 A031361 A031362 * A031364 A031365 A031366

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Erich Friedman

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

Status

approved