cp's OEIS Frontend

A387031 All integers k that can produce a closed walk in an equilateral triangular lattice via noncongruent primitive k-length diagonals, in ascending order.

%I A387031 #5 Aug 18 2025 16:01:39
%S A387031 53599,104377,105469,121303,126217,136591,144781,172081,177289,178087,
%T A387031 189007,205933,211603,222859,251503,273637,276241,290563,300181,
%U A387031 300979,307489,325717,345247,346801,348859,358267,359233,388759,392119,392977,403039,417487
%N A387031 All integers k that can produce a closed walk in an equilateral triangular lattice via noncongruent primitive k-length diagonals, in ascending order.
%C A387031 All observed terms are products of 4 different primes that are 1 mod 6 (A002476), though not all such products produce closed walks. It is conjectured that all terms are products of 4 or more such primes, including at least 4 different ones.
%C A387031 Closed walks along diagonals in a square lattice gives A386671.
%e A387031 a(1) = 53599 because 53599-length moves of [0, 60, 120] degrees respectively of [26216, 35445, 0] + [-16165, 0, 43656] + [-19651, 0, 41000] + [-1389, -52891, 0] + [0, -21829, -39240] + [50264, 0, -6141] = [39275, -39275, 39275], and [0, 0] + 39275 @ 0 degrees - 39275 @ 60 degrees + 39275 @ 120 degrees = [0, 0].
%o A387031 (PARI) is_a387031(k)={my(v=List); for(x=1, sqrtint(k^2/3), my(y=(sqrtint(4*k^2-3*x^2)-x)/2); if((x+y/2)^2+y^2*3/4==k^2 && gcd([x, y, k])==1, listput(v, [x, y]))); return(if(#v>=3 && closable(v), 1, 0))}
%o A387031 closable(v, c=vector(4))={my(o=(c[1]==c[3] && c[2]==-c[3])); if(!#v, return(if(c[4], o, 0))); my(x, y, v2=v); listpop(v2); foreach(if(o, [0, 1], [0, 1, 2, 3, 4, 5, 6]), r1, my(r2=r1%6+1); forperm(2, p, my(c2=c); if(r1, c2[(r1-1)%3+1]+=v[#v][p[1]]*if(r1>=4, -1, 1); c2[(r2-1)%3+1]+=v[#v][p[2]]*if(r2>=4, -1, 1); c2[4]++); if(closable(v2, c2), return(1)); if(!r1, break)))}
%Y A387031 Cf. A002476, A386671.
%K A387031 nonn
%O A387031 1,1
%A A387031 _Charles L. Hohn_, Aug 13 2025