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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

Colin Barker, Table of n, a(n) for n = 0..1000

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

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

Cf. A008579, A298802.

Sequence in context: A049305 A147606 A279083 * A085147 A298810 A298811

Adjacent sequences:  A298802 A298803 A298804 * A298806 A298807 A298808

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified January 19 15:37 EST 2020. Contains 331049 sequences. (Running on oeis4.)