A298803 Growth series for group with presentation < S, T : S^3 = T^3 = (S*T)^4 = 1 >.

1, 4, 8, 16, 30, 50, 88, 150, 260, 448, 768, 1328, 2284, 3930, 6776, 11662, 20082, 34592, 59560, 102570, 176642, 304180, 523830, 902084, 1553452, 2675184, 4606892, 7933444, 13662066, 23527220, 40515838, 69771678, 120152672, 206912968, 356321478, 613615442

Offset: 0

Author

John Cannon and N. J. A. Sloane, Feb 04 2018

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,1,3,1,0,-1).

Crossrefs

Cf. A008579, A298802.

Programs

Magma

R := RationalFunctionField(Integers());
PSR25 := PowerSeriesRing(Integers():Precision := 25);
FG := FreeGroup(2);
TG := quo;
f, A :=IsAutomaticGroup(TG);
gf := GrowthFunction(A);
R!gf;
Coefficients(PSR25!gf);

Mathematica

LinearRecurrence[{0,1,3,1,0,-1},{1,4,8,16,30,50,88,150},40] (* Harvey P. Dale, May 03 2019 *)

PARI

Vec((1 + 4*x + 7*x^2 + 9*x^3 + 9*x^4 + 6*x^5 + 3*x^6 - 2*x^7) / ((1 + x + x^2)*(1 - x - x^2 - x^3 + x^4)) + O(x^40)) \\ Colin Barker, Feb 04 2018

Formula

G.f.: (1 + 4*x + 7*x^2 + 9*x^3 + 9*x^4 + 6*x^5 + 3*x^6 - 2*x^7) / ((1 + x + x^2)*(1 - x - x^2 - x^3 + x^4)). [Corrected by Colin Barker, Feb 04 2018]

a(n) = a(n-2) + 3*a(n-3) + a(n-4) - a(n-6) for n>7. - Colin Barker, Feb 04 2018