A396927 - OEIS

%I #26 Jun 22 2026 11:10:48

%S 1,6,20,54,136,334,812,1966,4752,11478,27716,66918,161560,390046,

%T 941660,2273374,5488416,13250214,31988852,77227926,186444712,

%U 450117358,1086679436,2623476238,6333631920,15290740086,36915112100,89120964294,215157040696,519435045694,1254027132092

%N Breadth-first search (BFS) layer sizes of the Coxeter group of the 5-dimensional right-corner orthoscheme with five pairwise distinct leg lengths.

%C 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.

%C 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.

%C 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.

%H Paolo Xausa, <a href="/A396927/b396927.txt">Table of n, a(n) for n = 0..1000</a>

%H Vico Bonfioli, <a href="https://github.com/elvec1o/pythagorean-reflection-sequence">The Universal Right-Triangle Reflection Sequence (Unequal Legs)</a>, GitHub repository, 2026.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1).

%F G.f.: (1 + x)^3 / ((1 - x)*(1 - 2*x - x^2)).

%F 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.

%F 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)).

%e a(1) = 6: the six reflections in the bounding facets are the 6 elements of word-length 1.

%t LinearRecurrence[{3, -1, -1}, {1, 6, 20, 54}, 35] (* _Paolo Xausa_, Jun 22 2026 *)

%o (Python)

%o def a(n):

%o     seq = [1, 6, 20, 54]

%o     while len(seq) <= n:

%o         seq.append(3*seq[-1] - seq[-2] - seq[-3])

%o     return seq[n]

%o print([a(n) for n in range(30)])

%Y Cf. A000129 (Pell numbers; the ratio a(n+1)/a(n) -> 1 + sqrt(2) is the Pell ratio).

%Y Cf. A005409, A293004 (same linear recurrence, signature (3,-1,-1)).

%Y Cf. A396406 (2D right-triangle reflection sequence of this project).

%K nonn,easy

%O 0,2

%A _Vico Bonfioli_, Jun 10 2026

%E More terms from _Paolo Xausa_, Jun 22 2026