cp's OEIS Frontend

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.

%I A357302 #29 Oct 02 2022 18:12:13
%S A357302 1,7,49,91,637,1729,12103,53599,375193,1983163,13882141,85276009,
%T A357302 596932063,4178524441,5201836549,36412855843,254889990901,
%U A357302 348523048783,2439661341481,17077629390367,25442182561159,178095277928113,1246666945496791,2009932422331561,14069526956320927
%N 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.
%C A357302 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.
%e A357302 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.
%e A357302 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].
%e A357302 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].
%e A357302 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.
%e A357302 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].
%e A357302 There are no squares with r > 18 between 49 and 90.
%e A357302 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].
%o A357302 (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))};
%o A357302 a357302(400000)
%o A357302 (PARI) \\ more efficient using function list_A344473 (see there)
%o A357302 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))};
%o A357302 a355703(17)
%o A357302 (Python)
%o A357302 from itertools import count, islice
%o A357302 from sympy.abc import x,y
%o A357302 from sympy.solvers.diophantine.diophantine import diop_quadratic
%o A357302 def A357302_gen(): # generator of terms
%o A357302     c = 0
%o A357302     for k in count(1):
%o A357302         if (d:=len(diop_quadratic(x*(x+y)+y**2-k**2))) > c:
%o A357302             yield k
%o A357302             c = d
%o A357302 A357302_list = print(list(islice(A357302_gen(),6))) # _Chai Wah Wu_, Sep 26 2022
%Y A357302 Cf. A002324, A003136, A004016, A050931, A088534, A230655, A357303.
%Y A357302 Cf. A087503, A246360.
%K A357302 nonn
%O A357302 1,2
%A A357302 _Hugo Pfoertner_, Sep 25 2022