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 10 results.

A250122 Coordination sequence for planar net 3.12.12.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 14, 15, 18, 21, 22, 24, 28, 30, 30, 33, 38, 39, 38, 42, 48, 48, 46, 51, 58, 57, 54, 60, 68, 66, 62, 69, 78, 75, 70, 78, 88, 84, 78, 87, 98, 93, 86, 96, 108, 102, 94, 105, 118, 111, 102, 114, 128, 120, 110, 123, 138, 129
Offset: 0

Views

Author

Darrah Chavey, Nov 23 2014

Keywords

Comments

Also, growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^6 = 1 >. See Magma program in A298805. - N. J. A. Sloane, Feb 06 2018

Crossrefs

List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).
Cf. A298805.

Programs

  • Mathematica
    Join[{1, 3, 4}, LinearRecurrence[{2, -3, 4, -3, 2, -1}, {6, 8, 12, 14, 15, 18}, 100]] (* Jean-François Alcover, Aug 05 2018 *)

Formula

From Joseph Myers, Nov 28 2014: (Start)
Empirically,
a(4n) = 10n - 2 except for a(0) = 1
a(4n+1) = 9n + 3
a(4n+2) = 8n + 6 except for a(2) = 4
a(4n+3) = 9n + 6. (End)
If these are correct, the sequence has g.f.
-(-1 - x - x^2 - 3*x^3 + x^4 - 5*x^5 + 3*x^6 - 4*x^7 + 2*x^8)/((x - 1)^2*(x^2 + 1)^2). - N. J. A. Sloane, Nov 28 2014
All the above conjectures are true. - N. J. A. Sloane, Dec 31 2015
E.g.f.: (9*x*cosh(x) - 4*(2*cos(x) + x^2 - 3) + 9*x*sinh(x) - (x - 3)*sin(x))/4. - Stefano Spezia, Jan 05 2023

Extensions

a(8) onwards from Maurizio Paolini and Joseph Myers (independently), Nov 28 2014

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

Crossrefs

Programs

  • Magma
    // See Magma program in A298805.

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

Crossrefs

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

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

Crossrefs

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

Crossrefs

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

Crossrefs

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

Crossrefs

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

Crossrefs

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.

Crossrefs

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

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

Original entry on oeis.org

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

Views

Author

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

Keywords

Crossrefs

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 + 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

Formula

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
Showing 1-10 of 10 results.