A334654 Number of equilateral triangles of edge length square root of 2n and having vertices in Z^4, one of which is the origin.

96, 96, 672, 96, 1152, 672, 1344, 96, 3264, 1152, 2304, 672, 2496, 1344, 8064, 96, 3456, 3264, 3648, 1152, 9408, 2304, 4608, 672, 9312, 2496, 13632, 1344, 5760, 8064, 5952, 96, 16128, 3456, 16128, 3264, 7104, 3648, 17472, 1152, 8064, 9408, 8256, 2304, 39168, 4608

Offset

1,1

Comments

A parity argument shows that the edge length of an equilateral triangle with vertices in Z^4 must be the square root of an even integer.

A characterization of the planes in which an equilateral triangle with vertices in Z^4 can lie is given in the Ionascu reference.

Links

Table of n, a(n) for n=1..46.

E.J. Ionascu, Equilateral triangles in Z^4, arXiv:1209.0147 [math.NT], 2012-2013; Vietnam J. Math. 43 (3) (2015), 525-539.

Formula

a(2n) = a(n).

Mathematica

a[n_] := a[n] = If[ EvenQ[n], a[n/2], Block[{p, c=0, v = Tuples[ {1, -1}, 4]}, p = Union@ Flatten[ Table[ Union[ Permutations /@ ((q #) & /@ v)], {q, PowersRepresentations[2 n, 4, 2]}], 2]; Do[ If[ Total[ (p[[i]] - p[[j]])^2] == 2 n, c++], {i, Length@ p}, {j, i-1}]; c]]; Array[a, 30] (* Giovanni Resta, May 08 2020 *)

Crossrefs

Sequence in context: A252715 A050277 A090221 * A045528 A181470 A306104

Adjacent sequences: A334651 A334652 A334653 * A334655 A334656 A334657

Keyword

nonn

Author

Matt Noble and Will Farran, May 07 2020

Status

approved