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

Author

Henry L. Fleischmann, Aug 04 2021

Keywords

• nonn
• easy

Comments

Proved via a combinatorial argument.

Examples

			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.
		

Links

• 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).

Crossrefs

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

Programs

Python

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

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