A346796 Number of equivalence classes of triangles in an n-dimensional hypercube, equivalent up to translation of difference vectors corresponding to edges.

0, 2, 22, 180, 1340, 9622, 68082, 478760, 3357880, 23524842, 164732942, 1153307740, 8073685620, 56517393662, 395626538602, 2769400119120, 19385843880560, 135701036304082, 949907641549062, 6649354653104900

Offset

1,2

Comments

Proved via a combinatorial argument.

Links

Table of n, a(n) for n=1..20.

Henry L. Fleischmann et al., Distinct Angle Problems and Variants, arXiv:2108.12015 [math.CO], 2021.

Index entries for linear recurrences with constant coefficients, signature (11,-31,21).

Formula

a(n) = (7^n - 3^(n+1) + 2)/12.

a(n) = 2*A016212(n-2) for n >= 2.

G.f.: 2*x^2/(1 - 11*x + 31*x^2 - 21*x^3). - Stefano Spezia, Aug 04 2021

Example

The 1-dimensional hypercube (vertices 0 and 1 on a line) has no triangles and thus no classes of triangle equivalent up to edge translation, so a(1)=0.
A square, the 2-dimensional hypercube, has two distinct right triangles up to edge translation, so a(2)=2.

Prog

(Python) def a(n): return (7**n - 3**(n+1) + 2)//12

Crossrefs

Cf. A016212 (allowing flips as well as edge translations, up to offset).

Sequence in context: A270407 A000184 A007613 * A279801 A043037 A058441

Adjacent sequences: A346793 A346794 A346795 * A346797 A346798 A346799

Keyword

nonn,easy

Author

Henry L. Fleischmann, Aug 04 2021

Status

approved