A385525
Consider the graph whose vertices are the points of the n-dimensional cubic lattice with points connected by all integer-length diagonals that traverse all n dimensions and do not intersect intermediate points. a(n) is the total length of the shortest possible closed walk in this graph via noncongruent diagonals of the same length.
Original entry on oeis.org
327080, 84, 52, 32, 18, 24, 24, 24, 24, 24, 18, 24, 24, 24, 24, 24, 24, 24, 24, 30, 24, 30, 30, 24, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 36, 30, 36, 36, 30, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 40, 36, 36, 40, 36, 40, 42
Offset: 2
a(2) = 327080 because [3636, 40723] + [8844, 39917] + [11603, -39204] + [38076, -14893] + [-37523, -16236] + [35844, -19667] + [-34387, -22116] + [-26093, 31476] = [0, 0] and 8 segments * length 40885 = 327080, which is the smallest example for n = 2.
a(3) = 84: [16, 11, 8] + [-13, 4, 16] + [-8, -19, -4] + [5, 4, -20] = [0, 0, 0] and 4 * 21 = 84.
a(4) = 52: [8, 8, 5, 4] + [-9, -6, 6, 4] + [-7, -4, -10, 2] + [8, 2, -1, -10] = [0, 0, 0, 0] and 4 * 13 = 52.
Cf.
A020882 (diagonals in 2 dimensions),
A096910 (diagonals in 3 dimensions).
A387031
All integers k that can produce a closed walk in an equilateral triangular lattice via noncongruent primitive k-length diagonals, in ascending order.
Original entry on oeis.org
53599, 104377, 105469, 121303, 126217, 136591, 144781, 172081, 177289, 178087, 189007, 205933, 211603, 222859, 251503, 273637, 276241, 290563, 300181, 300979, 307489, 325717, 345247, 346801, 348859, 358267, 359233, 388759, 392119, 392977, 403039, 417487
Offset: 1
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].
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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))}
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)))}
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