OFFSET
0,2
COMMENTS
Growth series for group with presentation < X, Y, Z | X^2 = Y^2, X^2 = Z^2, X^2 = (Y*Z)^3, X^2 = (Z*X)^2, X^2 = (X*Y)^6 >. Probably Shutov intended to add "X^2 = Id" to the presentation, which would have produced the sequence A072154.
LINKS
A. V. Shutov, On the number of words of a given length in plane crystallographic groups (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 302 (2003), Anal. Teor. Chisel i Teor. Funkts. 19, 188--197, 203. See Table 1, line "p6m".
A. V. Shutov, On the number of words of a given length in plane crystallographic groups (English translation), J. Math. Sci. (N.Y.) 129 (2005), no. 3, 3922-3926 [MR2023041]. See Table 1, line "p6m".
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).
FORMULA
From Bruno Berselli, Apr 09 2018: (Start)
G.f.: (x + 1)*(x^7 - 2*x^6 + 3*x^5 + 2*x^4 + 2*x^3 + x^2 + 4*x + 1)/((x - 1)^2*(x^4 + x^3 + x^2 + x + 1)).
a(5*k) = 24*k with k>0, a(0)=1;
a(5*k+1) = 24*k + 6;
a(5*k+2) = 24*k + 10 with k>0, a(2)=11;
a(5*k+3) = 24*k + 14;
a(5*k+4) = 24*k + 18. (End)
PROG
(Magma)
R<x> := RationalFunctionField(Integers());
FG3<X, Y, Z> := FreeGroup(3);
Q3 := quo<FG3| X^2=Y^2, X^2=Z^2, X^2 = (Y*Z)^3, X^2 = (Z*X)^2, X^2 = (X*Y)^6 >;
G3 := AutomaticGroup(Q3);
f3 := GrowthFunction(G3);
R!f3;
PSR := PowerSeriesRing(Integers():Precision := 60);
Coefficients(PSR!f3);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Mar 30 2018
STATUS
approved