OFFSET
0,2
COMMENTS
The 5D right-corner orthoscheme has vertex V_0 with five mutually perpendicular legs of pairwise distinct positive lengths. W is the reflection group generated by reflections in the six bounding facets; a(n) is the number of distinct group elements of minimal W-word-length n.
a(n) equals the coefficients of the exact Coxeter (Steinberg-Solomon) Poincaré growth series of this diagram: a path on six nodes whose consecutive facet pairs are labeled infinity and whose non-consecutive pairs meet perpendicularly (m = 2, commuting). This is not the free Coxeter maximum, since the perpendicular pairs commute.
Leg-independence (Class C, all n >= 3): for pairwise distinct legs every consecutive dihedral angle is an irrational multiple of Pi (Niven-Conway-Jones), which reduces the relevant relation-finding to rank-0 descent on specific elliptic curves; hence the affine representation is faithful, the realized geometric growth equals the abstract Coxeter word-growth, and a(n) is a leg-independent closed form (Bonfioli 2026, Class C). Confirmed by exact-rational breadth-first search (BFS) on legs (1, 2, 3, 5, 7) through depth 12.
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..1000
Vico Bonfioli, The Universal Right-Triangle Reflection Sequence (Unequal Legs), GitHub repository, 2026.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).
FORMULA
G.f.: (1 + x)^3 / ((1 - x)*(1 - 2*x - x^2)).
a(0)=1, a(1)=6, a(2)=20, a(3)=54; a(n) = 3*a(n-1) - a(n-2) - a(n-3) for n >= 4.
a(n+1)/a(n) -> 1 + sqrt(2) = 2.41421356..., the largest root of x^3 - 3*x^2 + x + 1 = (x-1)*(x^2-2*x-1) (the reverse of the g.f. denominator). Equivalently 1 + 2*cos(2*Pi/8), the n = 5 case of r_n = 1 + 2*cos(2*Pi/(n+3)).
EXAMPLE
a(1) = 6: the six reflections in the bounding facets are the 6 elements of word-length 1.
MATHEMATICA
LinearRecurrence[{3, -1, -1}, {1, 6, 20, 54}, 35] (* Paolo Xausa, Jun 22 2026 *)
PROG
(Python)
def a(n):
seq = [1, 6, 20, 54]
while len(seq) <= n:
seq.append(3*seq[-1] - seq[-2] - seq[-3])
return seq[n]
print([a(n) for n in range(30)])
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vico Bonfioli, Jun 10 2026
EXTENSIONS
More terms from Paolo Xausa, Jun 22 2026
STATUS
approved