|
| |
|
|
A298805
|
|
Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^7 = 1 >.
|
|
11
|
|
|
|
1, 3, 4, 6, 8, 12, 16, 22, 24, 34, 40, 56, 62, 83, 98, 133, 152, 202, 236, 322, 368, 496, 570, 776, 892, 1202, 1384, 1871, 2158, 2915, 3352, 4534, 5218, 7060, 8120, 10976, 12636, 17084, 19664, 26580, 30592, 41367, 47604, 64365, 74072, 100152, 115264, 155836, 179352, 242488, 279076, 377324, 434246, 587126
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
Index entries for linear recurrences with constant coefficients, signature (-1,0,0,1,2,1,0,1,0,1,2,1,0,0,-1,-1).
|
|
|
FORMULA
|
G.f.: (-2*x^18 - 2*x^17 + 3*x^16 + 6*x^15 + 9*x^14 + 12*x^13 + 15*x^12 + 19*x^11 + 21*x^10 + 21*x^9 + 21*x^8 + 21*x^7 + 17*x^6 + 15*x^5 + 13*x^4 + 10*x^3 + 7*x^2 + 4*x + 1)/(x^16 + x^15 - x^12 - 2*x^11 - x^10 - x^8 - x^6 - 2*x^5 - x^4 + x + 1).
The denominator can be factored: G.f. also = -(2*x^18 + 2*x^17 - 3*x^16 - 6*x^15 - 9*x^14 - 12*x^13 - 15*x^12 - 19*x^11 - 21*x^10 - 21*x^9 - 21*x^8 - 21*x^7 - 17*x^6 - 15*x^5 - 13*x^4 - 10*x^3 - 7*x^2 - 4*x - 1) / ((x^4 + x^3 + x^2 + x + 1) * (x^12 - x^10 - x^8 + x^6 - x^4 - x^2 + 1)).
a(n) = -a(n-1) + a(n-4) + 2*a(n-5) + a(n-6) + a(n-8) + a(n-10) + 2*a(n-11) + a(n-12) - a(n-15) - a(n-16) for n>18. - Colin Barker, Feb 06 2018
|
|
|
MATHEMATICA
|
LinearRecurrence[{-1, 0, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 0, -1, -1}, {1, 3, 4, 6, 8, 12, 16, 22, 24, 34, 40, 56, 62, 83, 98, 133, 152, 202, 236}, 60] (* Harvey P. Dale, Jun 15 2021 *)
|
|
|
PROG
|
(Magma)
// To get the growth function for the group with presentation
// < S, T | S^a = T^b = (S*I)^c = 1 >
a:=2; b:=3; c:=7;
R<x> := RationalFunctionField(Integers());
PSR := PowerSeriesRing(Integers():Precision := 100);
FG<S, T> := FreeGroup(2);
TG := quo<FG | S^a, T^b, (S*T)^c >;
f, A :=IsAutomaticGroup(TG);
gf := GrowthFunction(A);
R!gf;
Coefficients(PSR!gf);
(PARI) Vec((1 + 4*x + 7*x^2 + 10*x^3 + 13*x^4 + 15*x^5 + 17*x^6 + 21*x^7 + 21*x^8 + 21*x^9 + 21*x^10 + 19*x^11 + 15*x^12 + 12*x^13 + 9*x^14 + 6*x^15 + 3*x^16 - 2*x^17 - 2*x^18) / ((1 + x + x^2 + x^3 + x^4)*(1 - x^2 - x^4 + x^6 - x^8 - x^10 + x^12)) + O(x^60)) \\ Colin Barker, Feb 06 2018
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
