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a(n) is the number of irreducible finite Coxeter groups in n dimensions, or -1 if there are an infinite number.
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%I #21 Jul 23 2024 10:50:01

%S 1,-1,3,5,3,4,4,4,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,

%T 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,

%U 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3

%N a(n) is the number of irreducible finite Coxeter groups in n dimensions, or -1 if there are an infinite number.

%C For n > 8, the Coxeter groups are exactly A(n), B(n) = C(n), and D(n), hence a(n) = 3.

%D H. S. M. Coxeter, Regular Polytopes, Dover Publications, Inc., 1973.

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

%F G.f.: (1 - 2*x + 4*x^2 + 2*x^3 - 2*x^4 + x^5 - x^8)/(1 - x). - _Stefano Spezia_, Jul 15 2024

%e For n = 4, there are five finite groups, denoted A(4) (symmetry group of the simplex), B(4) (= C(4)) (symmetry group of the tesseract and the 4-dimensional cross polytope), D(4) (symmetry group of the demitesseract), F(4) (symmetry group of the 24-cell) and H(4) (symmetry group of the 120-cell and the 600-cell).

%o (PARI) a(n)=if(n>8,3,[1,-1,3,5,3,4,4,4][n]) \\ _Charles R Greathouse IV_, Jul 15 2024

%Y Cf. A060296, A358241.

%K easy,sign

%O 1,3

%A _Douglas Boffey_, Jul 14 2024