cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

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

Views

Author

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

Keywords

Comments

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

Links

Crossrefs

Cf. A008579, A298802, A298805.

Programs

  • Magma
    // See Magma program in A298805.

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

Views

Author

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

Keywords

Links

Crossrefs

Cf. A008579, A298802.

Programs

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

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

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

Views

Author

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

Keywords

Links

Crossrefs

Cf. A008579, A298802, A298805.

Programs

  • Magma
    // See Magma program in A298805.
  • PARI
    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

Formula

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

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

Original entry on oeis.org

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

Views

Author

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

Keywords

Links

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

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

Views

Author

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

Keywords

Links

Crossrefs

Cf. A008579, A298802, A298805.

Programs

  • Magma
    // See Magma program in A298805.
  • PARI
    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

Formula

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

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

Views

Author

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

Keywords

Links

Crossrefs

Cf. A008579, A298802, A298805.

Programs

  • Magma
    // See Magma program in A298805.
  • Mathematica
    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 *)
  • PARI
    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

Formula

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

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

Views

Author

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

Keywords

Comments

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

Links

Crossrefs

Cf. A008579, A298802, A298805.

Programs

  • Magma
    // See Magma program in A298805.

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

Views

Author

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

Keywords

Links

Crossrefs

Cf. A008579, A298802, A298805.

Programs

  • Magma
    // See Magma program in A298805.
  • PARI
    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

Formula

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

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

Views

Author

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

Keywords

Links

Crossrefs

Cf. A008579, A298802, A298805.

Programs

  • Magma
    // See Magma program in A298805.
  • PARI
    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

Formula

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

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

Views

Author

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

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Magma
    // See Magma program in A298805.
  • PARI
    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

Formula

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
Showing 1-10 of 11 results. Next

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