A301730 Expansion of (x^8-x^7+x^6+5*x^5+4*x^4+3*x^3+5*x^2+5*x+1)/(x^6-x^5-x+1).

1, 6, 11, 14, 18, 24, 30, 34, 38, 42, 48, 54, 58, 62, 66, 72, 78, 82, 86, 90, 96, 102, 106, 110, 114, 120, 126, 130, 134, 138, 144, 150, 154, 158, 162, 168, 174, 178, 182, 186, 192, 198, 202, 206, 210, 216, 222, 226, 230, 234, 240, 246, 250, 254, 258, 264

Offset: 0

Author

N. J. A. Sloane, Mar 30 2018

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A072154.

Programs

Magma

R := RationalFunctionField(Integers());
FG3 := FreeGroup(3);
Q3 := quo;
G3 := AutomaticGroup(Q3);
f3 := GrowthFunction(G3);
R!f3;
PSR := PowerSeriesRing(Integers():Precision := 60);
Coefficients(PSR!f3);

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)