John Cannon - cp's OEIS frontend

A299253 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^12 = 1 >.

1, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 126, 183, 242, 357, 472, 696, 920, 1356, 1792, 2640, 3486, 5136, 6788, 10002, 13216, 19473, 25730, 37911, 50092, 73806, 97518, 143688, 189860, 279744, 369628, 544620, 719612, 1060296, 1400980, 2064243, 2727504, 4018785, 5310068, 7824000, 10337932, 15232200, 20126468

Offset: 0

John Cannon and N. J. A. Sloane, Feb 06 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,2,0,3,0,5,0,3,0,2,0,1,0,0,0,-1).

Cf. A008579, A298802, A298805.

Magma
```
// See Magma program in A298805.
```

Vec((1 + 3*x + 4*x^2 + 6*x^3 + 7*x^4 + 9*x^5 + 10*x^6 + 12*x^7 + 13*x^8 + 15*x^9 + 15*x^10 + 15*x^11 + 15*x^12 + 12*x^13 + 12*x^14 + 9*x^15 + 9*x^16 + 6*x^17 + 6*x^18 + 3*x^19 + 3*x^20 - 2*x^22) / ((1 - x + x^2 - x^3 + x^4)*(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

G.f.: (-2*x^22 + 3*x^20 + 3*x^19 + 6*x^18 + 6*x^17 + 9*x^16 + 9*x^15 + 12*x^14 + 12*x^13 + 15*x^12 + 15*x^11 + 15*x^10 + 15*x^9 + 13*x^8 + 12*x^7 + 10*x^6 + 9*x^5 + 7*x^4 + 6*x^3 + 4*x^2 + 3*x + 1)/(x^20 - x^16 - 2*x^14 - 3*x^12 - 5*x^10 - 3*x^8 - 2*x^6 - x^4 + 1).

a(n) = a(n-4) + 2*a(n-6) + 3*a(n-8) + 5*a(n-10) + 3*a(n-12) + 2*a(n-14) + a(n-16) - a(n-20) for n>20. - Colin Barker, Feb 06 2018

A299252 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^11 = 1 >.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 94, 120, 178, 232, 344, 448, 664, 864, 1280, 1662, 2459, 3202, 4741, 6168, 9132, 11880, 17588, 22880, 33870, 44068, 65246, 84880, 125664, 163484, 242036, 314880, 466176, 606478, 897892, 1168124, 1729394, 2249880, 3330929, 4333418, 6415591, 8346452, 12356856, 16075828, 23800132

Offset: 0

John Cannon and N. J. A. Sloane, Feb 06 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,0,0,1,1,2,2,3,4,3,2,3,2,3,2,3,2,3,4,3,2,2,1,1,0,0,-1,-1).

Cf. A008579, A298802, A298805.

Magma
```
// See Magma program in A298805.
```

Vec((1 + 4*x + 7*x^2 + 10*x^3 + 13*x^4 + 16*x^5 + 19*x^6 + 22*x^7 + 25*x^8 + 27*x^9 + 29*x^10 + 33*x^11 + 33*x^12 + 33*x^13 + 33*x^14 + 33*x^15 + 33*x^16 + 33*x^17 + 33*x^18 + 31*x^19 + 27*x^20 + 24*x^21 + 21*x^22 + 18*x^23 + 15*x^24 + 12*x^25 + 9*x^26 + 6*x^27 + 3*x^28 - 2*x^29 - 2*x^30) / ((1 + x + x^2)*(1 + x^3 + x^6)*(1 - x^2 - x^4 - x^6 - x^8 + x^10 - x^12 - x^14 - x^16 - x^18 + x^20)) + O(x^60)) \\ Colin Barker, Feb 06 2018

G.f.: (-2*x^30 - 2*x^29 + 3*x^28 + 6*x^27 + 9*x^26 + 12*x^25 + 15*x^24 + 18*x^23 + 21*x^22 + 24*x^21 + 27*x^20 + 31*x^19 + 33*x^18 + 33*x^17 + 33*x^16 + 33*x^15 + 33*x^14 + 33*x^13 + 33*x^12 + 33*x^11 + 29*x^10 + 27*x^9 + 25*x^8 + 22*x^7 + 19*x^6 + 16*x^5 + 13*x^4 + 10*x^3 + 7*x^2 + 4*x + 1)/(x^28 + x^27 - x^24 - x^23 - 2*x^22 - 2*x^21 - 3*x^20 - 4*x^19 - 3*x^18 - 2*x^17 - 3*x^16 - 2*x^15 - 3*x^14 - 2*x^13 - 3*x^12 - 2*x^11 - 3*x^10 - 4*x^9 - 3*x^8 - 2*x^7 - 2*x^6 - x^5 - x^4 + x + 1).

a(n) = -a(n-1) + a(n-4) + a(n-5) + 2*a(n-6) + 2*a(n-7) + 3*a(n-8) + 4*a(n-9) + 3*a(n-10) + 2*a(n-11) + 3*a(n-12) + 2*a(n-13) + 3*a(n-14) + 2*a(n-15) + 3*a(n-16) + 2*a(n-17) + 3*a(n-18) + 4*a(n-19) + 3*a(n-20) + 2*a(n-21) + 2*a(n-22) + a(n-23) + a(n-24) - a(n-27) - a(n-28) for n>30. - Colin Barker, Feb 06 2018

A298812 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^10 = 1 >.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 16, 24, 32, 48, 62, 87, 114, 165, 216, 312, 408, 588, 766, 1104, 1444, 2082, 2720, 3921, 5122, 7383, 9642, 13902, 18164, 26184, 34204, 49308, 64412, 92856, 121298, 174867, 228438, 329313, 430188, 620160, 810132, 1167888, 1525642, 2199372, 2873104, 4141866, 5410628, 7799973, 10189318, 14688939

Offset: 0

John Cannon and N. J. A. Sloane, Feb 06 2018

The initial coefficients for the group S, T : S^2 = T^3 = (S*T)^m = 1 > approach A029744 as m increases.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,2,0,4,0,2,0,1,0,0,0,-1).

Cf. A008579, A298802, A298805, A299252, A029744.

Magma
```
// See Magma program in A298805.
```

Vec((1 + 3*x + 4*x^2 + 6*x^3 + 7*x^4 + 9*x^5 + 10*x^6 + 12*x^7 + 12*x^8 + 12*x^9 + 12*x^10 + 9*x^11 + 9*x^12 + 6*x^13 + 6*x^14 + 3*x^15 + 3*x^16 - 2*x^18) / ((1 + x^2)^2*(1 + x^4)*(1 - 2*x^2 + x^4 - 2*x^6 + x^8)) + O(x^60)) \\ Colin Barker, Feb 06 2018

G.f.: (-2*x^18 + 3*x^16 + 3*x^15 + 6*x^14 + 6*x^13 + 9*x^12 + 9*x^11 + 12*x^10 + 12*x^9 + 12*x^8 + 12*x^7 + 10*x^6 + 9*x^5 + 7*x^4 + 6*x^3 + 4*x^2 + 3*x + 1)/(x^16 - x^12 - 2*x^10 - 4*x^8 - 2*x^6 - x^4 + 1).

a(n) = a(n-4) + 2*a(n-6) + 4*a(n-8) + 2*a(n-10) + a(n-12) - a(n-16) for n>16. - Colin Barker, Feb 06 2018

A298811 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^9 = 1 >.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 16, 24, 32, 46, 56, 82, 104, 152, 192, 280, 350, 507, 642, 933, 1176, 1708, 2152, 3122, 3940, 5726, 7216, 10480, 13212, 19188, 24190, 35140, 44300, 64338, 81112, 117809, 148522, 215717, 271960, 394998, 497972, 723268, 911828, 1324360, 1669626, 2425008, 3057212, 4440362, 5597988, 8130648

Offset: 0

John Cannon and N. J. A. Sloane, Feb 06 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,0,0,1,1,2,3,2,1,2,1,2,1,2,3,2,1,1,0,0,-1,-1).

Cf. A008579, A298802, A298805.

Magma
```
// See Magma program in A298805.
```

Vec((1 + 4*x + 7*x^2 + 10*x^3 + 13*x^4 + 16*x^5 + 19*x^6 + 21*x^7 + 23*x^8 + 27*x^9 + 27*x^10 + 27*x^11 + 27*x^12 + 27*x^13 + 27*x^14 + 25*x^15 + 21*x^16 + 18*x^17 + 15*x^18 + 12*x^19 + 9*x^20 + 6*x^21 + 3*x^22 - 2*x^23 - 2*x^24) / ((1 - x + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)*(1 - 2*x^2 + x^6 - 2*x^10 + x^12)) + O(x^60)) \\ Colin Barker, Feb 06 2018

G.f.: (-2*x^24 - 2*x^23 + 3*x^22 + 6*x^21 + 9*x^20 + 12*x^19 + 15*x^18 + 18*x^17 + 21*x^16 + 25*x^15 + 27*x^14 + 27*x^13 + 27*x^12 + 27*x^11 + 27*x^10 + 27*x^9 + 23*x^8 + 21*x^7 + 19*x^6 + 16*x^5 + 13*x^4 + 10*x^3 + 7*x^2 + 4*x + 1)/(x^22 + x^21 - x^18 - x^17 - 2*x^16 - 3*x^15 - 2*x^14 - x^13 - 2*x^12 - x^11 - 2*x^10 - x^9 - 2*x^8 - 3*x^7 - 2*x^6 - x^5 - x^4 + x + 1).

a(n) = -a(n-1) + a(n-4) + a(n-5) + 2*a(n-6) + 3*a(n-7) + 2*a(n-8) + a(n-9) + 2*a(n-10) + a(n-11) + 2*a(n-12) + a(n-13) + 2*a(n-14) + 3*a(n-15) + 2*a(n-16) + a(n-17) + a(n-18) - a(n-21) - a(n-22) for n>24. - Colin Barker, Feb 06 2018

A298810 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^8 = 1 >.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 16, 24, 30, 39, 50, 69, 88, 120, 150, 204, 260, 354, 448, 609, 768, 1047, 1328, 1806, 2284, 3108, 3930, 5352, 6776, 9219, 11662, 15873, 20082, 27336, 34592, 47076, 59560, 81066, 102570, 139605, 176642, 240411, 304180, 414006, 523830

Offset: 0

John Cannon and N. J. A. Sloane, Feb 06 2018

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

Cf. A008579, A298802, A298805.

Magma
```
// See Magma program in A298805.
```

Vec((1 + 3*x + 4*x^2 + 6*x^3 + 7*x^4 + 9*x^5 + 9*x^6 + 9*x^7 + 9*x^8 + 6*x^9 + 6*x^10 + 3*x^11 + 3*x^12 - 2*x^14) / ((1 - x + x^2)*(1 + x + x^2)*(1 - x^2 - x^4 - x^6 + x^8)) + O(x^60)) \\ Colin Barker, Feb 06 2018

G.f.: (-2*x^14 + 3*x^12 + 3*x^11 + 6*x^10 + 6*x^9 + 9*x^8 + 9*x^7 + 9*x^6 + 9*x^5 + 7*x^4 + 6*x^3 + 4*x^2 + 3*x + 1)/(x^12 - x^8 - 3*x^6 - x^4 +1).

a(n) = a(n-4) + 3*a(n-6) + a(n-8) - a(n-12) for n>12. - Colin Barker, Feb 06 2018

A298809 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^5 = 1 >.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 8, 10, 6, 3, 1

Offset: 0

John Cannon and N. J. A. Sloane, Feb 06 2018

This group is finite, so the growth series is a polynomial.

Coordination sequence for truncated dodecahedron (see Karzes link). - N. J. A. Sloane, Nov 20 2019

Tom Karzes, Polyhedron Coordination Sequences
N. J. A. Sloane, The 60 vertices of the truncated dodecahedron labeled with their distance from a base vertex.
Index entries for coordination sequences

Cf. A008579, A298802, A298805.

Magma
```
// See Magma program in A298805.
```

A298808 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^4 = 1 >.

Original entry on oeis.org

1, 3, 4, 6, 6, 3, 1

Offset: 0

John Cannon and N. J. A. Sloane, Feb 06 2018

This group is finite, so the growth series is a polynomial.

Coordination sequence for truncated cube (see Karzes link). The coordination sequence for the cuboctahedron is 1,4,6,1, which is too short to have its own entry. - N. J. A. Sloane, Nov 20 2019

Tom Karzes, Polyhedron Coordination Sequences
Index entries for coordination sequences

Cf. A008579, A298802, A298805.

Magma
```
// See Magma program in A298805.
```

A298807 Growth series for group with presentation < S, T : S^3 = T^3 = (S*T)^6 = 1 >.

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 126, 242, 472, 920, 1792, 3486, 6788, 13216, 25730, 50092, 97518, 189860, 369628, 719612, 1400980, 2727504, 5310068, 10337932, 20126468, 39183340, 76284330, 148514636, 289136638, 562907480, 1095899956, 2133559698, 4153734080, 8086723216, 15743687792, 30650697262, 59672502090

Offset: 0

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

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,2,3,5,3,2,1,0,-1).

Cf. A008579, A298802, A298805.

Magma
```
// See Magma program in A298805.
```

LinearRecurrence[{0,1,2,3,5,3,2,1,0,-1},{1,4,8,16,32,64,126,242,472,920,1792,3486},40] (* Harvey P. Dale, Jul 02 2025 *)

Vec((1 + 4*x + 7*x^2 + 10*x^3 + 13*x^4 + 15*x^5 + 15*x^6 + 12*x^7 + 9*x^8 + 6*x^9 + 3*x^10 - 2*x^11) / ((1 + x + x^2 + x^3 + x^4)*(1 - x - x^2 - x^3 - x^4 - x^5 + x^6)) + O(x^40)) \\ Colin Barker, Feb 06 2018

G.f.: (-2*x^11 + 3*x^10 + 6*x^9 + 9*x^8 + 12*x^7 + 15*x^6 + 15*x^5 + 13*x^4 + 10*x^3 + 7*x^2 + 4*x + 1)/(x^10 - x^8 - 2*x^7 - 3*x^6 - 5*x^5 - 3*x^4 - 2*x^3 - x^2 + 1).

a(n) = a(n-2) + 2*a(n-3) + 3*a(n-4) + 5*a(n-5) + 3*a(n-6) + 2*a(n-7) + a(n-8) - a(n-10) for n>11. - Colin Barker, Feb 06 2018

A298806 Growth series for group with presentation < S, T : S^3 = T^6 = (S*T)^6 = 1 >.

Original entry on oeis.org

1, 4, 10, 25, 60, 148, 358, 869, 2106, 5110, 12396, 30070, 72942, 176939, 429214, 1041172, 2525640, 6126607, 14861710, 36051016, 87451296, 212136296, 514592810, 1248281249, 3028037016, 7345306340, 17817987338, 43222250797, 104847025002, 254334247970, 616955127612, 1496588180810, 3630371290710

Offset: 0

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

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,1,2,-3,2,1,-2,3,-1).

Cf. A008579, A298802, A298805.

Magma
```
// See Magma program in A298805.
```

Vec((1 - x + x^2)*(1 + x + x^2)*(1 + x - x^2 + x^3 - x^4 + x^5 + x^6) / (1 - 3*x + 2*x^2 - x^3 - 2*x^4 + 3*x^5 - 2*x^6 - x^7 + 2*x^8 - 3*x^9 + x^10) + O(x^40)) \\ Colin Barker, Feb 06 2018

G.f.: (x^10 + x^9 + 2*x^7 - x^6 + 3*x^5 - x^4 + 2*x^3 + x + 1)/(x^10 - 3*x^9 + 2*x^8 - x^7 - 2*x^6 + 3*x^5 - 2*x^4 - x^3 + 2*x^2 - 3*x + 1).

a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) + 2*a(n-4) - 3*a(n-5) + 2*a(n-6) + a(n-7) - 2*a(n-8) + 3*a(n-9) - a(n-10) for n>10. - Colin Barker, Feb 06 2018

A298805 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^7 = 1 >.

Original entry on oeis.org

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

Offset: 0

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

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).

Cf. A008579, A298802.

// 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 := RationalFunctionField(Integers());
PSR := PowerSeriesRing(Integers():Precision := 100);
FG := FreeGroup(2);
TG := quo;
f, A :=IsAutomaticGroup(TG);
gf := GrowthFunction(A);
R!gf;
Coefficients(PSR!gf);

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 *)

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

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

cp's OEIS Frontend

User: John Cannon

A299253 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^12 = 1 >.

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A299252 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^11 = 1 >.

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A298812 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^10 = 1 >.

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A298811 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^9 = 1 >.

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A298810 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^8 = 1 >.

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A298809 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^5 = 1 >.

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A298808 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^4 = 1 >.

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Magma

A298807 Growth series for group with presentation < S, T : S^3 = T^3 = (S*T)^6 = 1 >.

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