A357302 Numbers k such that k^2 can be represented as x^2 + x*y + y^2 in more ways than for any smaller k.

1, 7, 49, 91, 637, 1729, 12103, 53599, 375193, 1983163, 13882141, 85276009, 596932063, 4178524441, 5201836549, 36412855843, 254889990901, 348523048783, 2439661341481, 17077629390367, 25442182561159, 178095277928113, 1246666945496791, 2009932422331561, 14069526956320927

Offset

1,2

Comments

Apparently the number of grid points t(n) = {1, 2, 3, 5, 8, 14, 23, ...} (A357303) in the reduced representations as described in the examples matches t(n) = A087503(n-3) + 2 for n >= 3, i.e., t(n) = t(n-1) + 3*t(n-2) - 3*t(n-3) for n >= 5. This coincidence persists up to t(15) = 1823, but t(16) = 2553, whereas the recurrence predicts 3281, which is t(17). It seems that all of the terms generated by the recurrence also appear as record numbers of grid points. However, there are other record numbers in between, of which 2553 is the first occurrence.

Links

Table of n, a(n) for n=1..25.

Example

The essential information in the complete set of representations of a square a(n)^2 can be extracted by taking into account the symmetries of the triangular lattice. If r is the number of all representations of a(n)^2, then there are t = (r/6 + 1)/2 pairs of triangular oblique coordinates lying in a sector of angular width Pi/6 completely containing the essential information.
a(1) = 1: r = 6 representations of 1^2 are [-1, 0], [-1, 1], [0, -1], [0, 1], [1, -1], [1, 0] reduced: (6/6 + 1)/2 = 1 grid point [1,0].
a(2) = 7: r = 18 representations of 7^2 = 49 are [-8, 5], [-7, 0], [-7, 7], [-5, -3], [-5, 8], [-3, -5], [-3, 8], [0, -7], [0, 7], [3, -8], [3, 5], [5, -8], [5, 3], [7, -7], [7, 0], [8, -5], [8, -3], [8, 3]; reduced: (18/6 + 1)/2 = 2 grid points [7, 0], [8, 3].
After a(2) = 7 there are no squares with more than 18 representations, e.g., r = 18 for 13^2, 14^2, 19^2, 21^2, ..., 42^2, 43^2.
a(3) = 49: r = 30 representations of 49^2 = 2401 are [-56, 21], [-56, 35], [-55, 16], [-55, 39], [-49, 0], [-49, 49], [-39, -16], [-39, 55], [-35, -21], [-35, 56], [-21, -35], [-21, 56], [-16, -39], [-16, 55], [0, -49], [0, 49], [16, -55], [16, 39], [21, -56], [21, 35], [35, -56], [35, 21], [39, -55], [39, 16], [49, -49], [49, 0], [55, -39], [55, -16], [56, -35], [56, -21]; reduced: (30/6 + 1)/2 = 3 grid points [49, 0], [55, 16], [56, 21].
There are no squares with r > 18 between 49 and 90.
a(4) = 91: r = 54 representations of 91^2 = 8281 are [-105,49], [-105,56], ..., [105, -56], [105,-49]; reduced: (54/6 + 1)/2 = 5 grid points [91, 0], [96, 11], [99, 19], [104, 39], [105, 49].

Prog

(PARI) a357302(upto) = {my (dmax=0); for (k = 1, upto, my (d = #qfbsolve (Qfb(1, 1, 1), k^2, 3)); if(d > dmax, print1(k, ", "); dmax=d))};
a357302(400000)
(PARI) \\ more efficient using function list_A344473 (see there)
a355703(maxexp10)= {my (sqterms=select(x->issquare(x), list_A344473 (10^(2*maxexp10))), r=0); for (k=1, #sqterms, my (d = #qfbsolve(Qfb(1, 1, 1), v[k], 3)); if (d>r, print1(sqrtint(v[k]), ", "); r=d))};
a355703(17)
(Python)
from itertools import count, islice
from sympy.abc import x, y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A357302_gen(): # generator of terms
    c = 0
    for k in count(1):
        if (d:=len(diop_quadratic(x*(x+y)+y**2-k**2))) > c:
            yield k
            c = d
A357302_list = print(list(islice(A357302_gen(), 6))) # Chai Wah Wu, Sep 26 2022

Crossrefs

Cf. A002324, A003136, A004016, A050931, A088534, A230655, A357303.

Cf. A087503, A246360.

Sequence in context: A343737 A344473 A230655 * A343771 A008461 A043131

Adjacent sequences: A357299 A357300 A357301 * A357303 A357304 A357305

Keyword

nonn

Author

Hugo Pfoertner, Sep 25 2022

Status

approved