OFFSET
0,2
COMMENTS
A four-dimensional orthoscheme (path simplex) is a simplex with vertices V_0,...,V_4 whose edges V_0V_1, V_1V_2, V_2V_3, V_3V_4 are mutually perpendicular; here the four edge lengths are distinct positive rationals. W is the group generated by the reflections in the five facets, and a(n) is the number of elements of W of word-length n in these reflections.
Two facets are perpendicular exactly when they are non-consecutive, so W is the right-angled Coxeter group with the path Coxeter diagram P_5; its growth series is given by Steinberg's formula, which yields the generating function and recurrence below.
The growth depends only on the perpendicularity pattern, not on the edge lengths. For dimension >= 3 the geometric orbit count equals this abstract growth for any distinct edge lengths (proved in the linked paper, using that distinct lengths force the consecutive dihedral angles irrational). The 2-dimensional analog is the right-triangle reflection group A396406. Verified by direct rational-arithmetic computation on edge lengths (1,2,3,5) through word-length 10.
LINKS
Vico Bonfioli, The Universal Right-Triangle Reflection Sequence (Unequal Legs), GitHub repository, 2026.
Vico Bonfioli, Data and proofs (Zenodo), doi:10.5281/zenodo.20370090, 2026.
Vico Bonfioli, Orthoscheme growth and Class C faithfulness, 2026.
Index entries for linear recurrences with constant coefficients, signature (2,1,-1).
FORMULA
G.f.: (1 + x)^3 / (1 - 2*x - x^2 + x^3).
a(0)=1, a(1)=5, a(2)=14, a(3)=33; a(n) = 2*a(n-1) + a(n-2) - a(n-3) for n >= 4.
a(n+1)/a(n) -> 1 + 2*cos(2*Pi/7) = 2.2469796..., the largest real root of x^3 - 2*x^2 - x + 1.
EXAMPLE
a(1) = 5: the five reflections in the bounding faces are the 5 elements of word-length 1.
MATHEMATICA
LinearRecurrence[{2, 1, -1}, {1, 5, 14, 33}, 33]
PROG
(Python)
def aseq(N):
s = [1, 5, 14, 33]
while len(s) < N:
s.append(2*s[-1] + s[-2] - s[-3])
return s
print(aseq(33))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vico Bonfioli, Jun 25 2026
STATUS
approved