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.

Previous Showing 11-20 of 49 results. Next

A078042 Expansion of (1-x)/(1+x-x^2+x^3).

Original entry on oeis.org

1, -2, 3, -6, 11, -20, 37, -68, 125, -230, 423, -778, 1431, -2632, 4841, -8904, 16377, -30122, 55403, -101902, 187427, -344732, 634061, -1166220, 2145013, -3945294, 7256527, -13346834, 24548655, -45152016, 83047505, -152748176, 280947697, -516743378, 950439251, -1748130326, 3215312955
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Comments

Absolute values give coordination sequence for (3,infinity,infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
a(n) is the upper left entry of the n-th power of the 3 X 3 matrix M = [-2, -2, 1; 1, 1, 0; 1, 0, 0]; a(n) = M^n [1, 1]. - Philippe Deléham, Apr 19 2023

Links

Crossrefs

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

Programs

  • Magma
    [n le 3 select -n*(-1)^n else -Self(n-1)+Self(n-2)-Self(n-3): n in [1..50]]; // Vincenzo Librandi, Dec 30 2015
  • Mathematica
    CoefficientList[Series[(1-x)/(1+x-x^2+x^3),{x,0,40}],x] (* or *) LinearRecurrence[{-1,1,-1},{1,-2,3},40] (* Harvey P. Dale, Jun 01 2012 *)
  • PARI
    Vec((1-x)/(1+x-x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = -a(n-1) + a(n-2) - a(n-3) for n > 2; a(0)=1, a(1)=-2, a(2)=3. - Harvey P. Dale, Jun 01 2012
a(n) = (-1)^n * A001590(n+2).
a(n) = Sum_{k=0..n} A188316(n,k)*(-2)^k. - Philippe Deléham, Apr 19 2023

A163876 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

Original entry on oeis.org

1, 3, 6, 12, 24, 48, 93, 180, 351, 684, 1332, 2592, 5046, 9825, 19128, 37239, 72498, 141144, 274788, 534972, 1041513, 2027676, 3947595, 7685400, 14962368, 29129580, 56711106, 110408373, 214949232, 418475259, 814711182, 1586125572, 3087958512
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Dec 03 2009

Keywords

Comments

Also, coordination sequence for (6,6,6) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
The initial terms coincide with those of A003945, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Crossrefs

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^6)/(1-2*x+2*x^6-x^7) )); // G. C. Greubel, Apr 25 2019
  • Mathematica
    coxG[{6,1,-1,40}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Mar 22 2015 *)
CoefficientList[Series[(1+x)*(1-x^6)/(1-2*x+2*x^6-x^7), {x,0,40}], x] (* G. C. Greubel, Aug 06 2017, modified Apr 25 2019 *)
  • PARI
    x='x+O('x^40); Vec((x^6+2*x^5+2*x^4+2*x^3+2*x^2+2*x+1)/(x^6-x^5- x^4-x^3-x^2-x+1)) \\ G. C. Greubel, Aug 06 2017
  • Sage
    ((1+x)*(1-x^6)/(1-2*x+2*x^6-x^7)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019

Formula

G.f.: (x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 1)/(x^6 - x^5 - x^4 - x^3 - x^2 - x + 1).
G.f.: (1+x)*(1-x^6)/(1-2*x+2*x^6-x^7). - G. C. Greubel, Apr 25 2019
a(n) = -a(n-6) + Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 07 2021

A265057 Coordination sequence for (2,3,7) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 16, 20, 24, 28, 33, 40, 48, 57, 67, 78, 92, 109, 129, 152, 178, 209, 246, 290, 342, 402, 472, 555, 653, 769, 905, 1064, 1251, 1471, 1731, 2037, 2396, 2818, 3314, 3898, 4586, 5395, 6346, 7464, 8779, 10327, 12148, 14290, 16809, 19771, 23256, 27356, 32179, 37852, 44524, 52372
Offset: 0

Views

Author

N. J. A. Sloane, Dec 29 2015

Keywords

Links

Crossrefs

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

Programs

  • Mathematica
    CoefficientList[Series[(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) (x^2 + x + 1) (x + 1)^2/(x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1), {x, 0, 60}], x] (* Vincenzo Librandi, Dec 30 2015 *)
  • PARI
    x='x+O('x^50); Vec((x^6+x^5+x^4+x^3+x^2+x+1)*(x^2+x+1)*(x+1)^2/(x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1)) \\ G. C. Greubel, Aug 06 2017

Formula

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

A265058 Coordination sequence for (2,3,8) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 16, 21, 27, 33, 40, 49, 61, 76, 94, 116, 142, 174, 214, 264, 326, 401, 493, 606, 745, 917, 1129, 1390, 1710, 2103, 2587, 3183, 3917, 4820, 5931, 7297, 8977, 11045, 13590, 16722, 20575, 25315, 31147, 38322, 47151, 58015, 71382, 87828, 108062, 132958, 163590, 201280, 247654
Offset: 0

Views

Author

N. J. A. Sloane, Dec 29 2015

Keywords

Links

Crossrefs

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

Programs

  • Mathematica
    CoefficientList[Series[(x + 1)^2 (x^2 + x + 1) (x^6 + x^4 + x^2 + 1)/(x^10 - x^7 - x^5 - x^3 + 1), {x, 0, 60}], x] (* Vincenzo Librandi, Dec 30 2015 *)
  • PARI
    x='x+O('x^50); Vec((x+1)^2*(x^2+x+1)*(x^6+x^4+x^2+1)/(x^10-x^7-x^5-x^3+1)) \\ G. C. Greubel, Aug 06 2017

Formula

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

A265059 Coordination sequence for (2,3,9) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 16, 21, 28, 36, 45, 56, 70, 89, 113, 143, 181, 228, 287, 361, 455, 575, 726, 916, 1155, 1456, 1836, 2315, 2920, 3684, 4647, 5861, 7391, 9321, 11756, 14827, 18701, 23587, 29749, 37520, 47320, 59681, 75272, 94936, 119737, 151016, 190466, 240221, 302973, 382119, 481941, 607840, 766627
Offset: 0

Views

Author

N. J. A. Sloane, Dec 29 2015

Keywords

Links

Crossrefs

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

Programs

  • Mathematica
    CoefficientList[Series[(x + 1)^2 (x^2 + x + 1) (x^6 + x^3 + 1)/(x^10 - x^8 - x^5 - x^2 + 1), {x, 0, 60}], x] (* Vincenzo Librandi, Dec 30 2015 *)
  • PARI
    x='x+O('x^50); Vec((x+1)^2*(x^2+x+1)*(x^6+x^3+1)/(x^10-x^8-x^5-x^2+1)) \\ G. C. Greubel, Aug 06 2017

Formula

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

A265060 Coordination sequence for (2,4,5) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 5, 8, 12, 16, 21, 28, 36, 46, 60, 77, 98, 126, 162, 207, 265, 340, 435, 557, 714, 914, 1170, 1499, 1920, 2458, 3148, 4032, 5163, 6612, 8468, 10844, 13887, 17785, 22776, 29167, 37353, 47836, 61260, 78452, 100469, 128664, 164772, 211014, 270232, 346069, 443190, 567566, 726846, 930827, 1192053, 1526588
Offset: 0

Views

Author

N. J. A. Sloane, Dec 29 2015

Keywords

Links

Crossrefs

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

Programs

  • Mathematica
    CoefficientList[Series[(x + 1)^2 (x^2 + 1) (x^4 + x^3 + x^2 + x + 1)/(x^8 - x^5 - x^4 - x^3 + 1), {x, 0, 60}], x] (* Vincenzo Librandi, Dec 30 2015 *)
  • PARI
    x='x+O('x^50); Vec((x+1)^2*(x^2+1)*(x^4+x^3+x^2+x+1)/(x^8-x^5-x^4-x^3+1)) \\ G. C. Greubel, Aug 06 2017

Formula

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

A265061 Coordination sequence for (2,4,6) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 5, 8, 12, 17, 24, 33, 45, 61, 83, 114, 155, 210, 286, 389, 529, 720, 979, 1331, 1810, 2462, 3349, 4554, 6193, 8423, 11455, 15579, 21188, 28815, 39188, 53296, 72483, 98577, 134064, 182327, 247965, 337232, 458636, 623745, 848292, 1153677, 1569001, 2133841, 2902023, 3946750, 5367579, 7299906, 9927870, 13501901
Offset: 0

Views

Author

N. J. A. Sloane, Dec 29 2015

Keywords

Links

Crossrefs

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

Programs

  • Mathematica
    CoefficientList[Series[(x + 1)^2 (x^2 + 1) (x^4 + x^2 + 1)/(x^8 - x^7 + x^6 - 2 x^5 + x^4 - 2 x^3 + x^2 - x + 1), {x, 0, 60}], x] (* Vincenzo Librandi, Dec 30 2015 *)
  • PARI
    Vec((x+1)^2*(x^2+1)*(x^4+x^2+1)/(x^8-x^7+x^6-2*x^5+x^4-2*x^3+x^2-x+1) + O(x^100)) \\ Altug Alkan, Dec 29 2015

Formula

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

A265062 Coordination sequence for (2,4,7) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 5, 8, 12, 17, 25, 36, 50, 70, 98, 137, 193, 271, 379, 531, 744, 1042, 1461, 2048, 2869, 4020, 5633, 7893, 11061, 15500, 21719, 30434, 42646, 59758, 83738, 117340, 164424, 230402, 322855, 452406, 633943, 888325, 1244781, 1744272, 2444193, 3424970, 4799303, 6725112, 9423686, 13205113, 18503907, 25928939, 36333403
Offset: 0

Views

Author

N. J. A. Sloane, Dec 29 2015

Keywords

Links

Crossrefs

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

Programs

  • Mathematica
    CoefficientList[Series[(x + 1)^2 (x^2 + 1) (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)/(x^10 - x^7 - x^6 - x^5 - x^4 - x^3 + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 31 2015 *)
LinearRecurrence[{0,0,1,1,1,1,1,0,0,-1},{1,3,5,8,12,17,25,36,50,70,98},50] (* Harvey P. Dale, May 17 2023 *)
  • PARI
    Vec((x+1)^2*(x^2+1)*(x^6+x^5+x^4+x^3+x^2+x+1)/(x^10-x^7-x^6-x^5-x^4-x^3+1) + O(x^50)) \\ Michel Marcus, Dec 31 2015

Formula

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

A265063 Coordination sequence for (2,4,8) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 5, 8, 12, 17, 25, 37, 53, 75, 107, 152, 216, 309, 441, 628, 895, 1275, 1816, 2588, 3689, 5257, 7491, 10675, 15211, 21675, 30888, 44016, 62723, 89381, 127368, 181499, 258637, 368560, 525200, 748413, 1066493, 1519757, 2165661, 3086079, 4397679, 6266716, 8930104, 12725445, 18133825, 25840796, 36823271, 52473355
Offset: 0

Views

Author

N. J. A. Sloane, Dec 29 2015

Keywords

Links

Crossrefs

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

Programs

  • Mathematica
    CoefficientList[Series[(x^2 + 1)*(x^4 + 1)*(x + 1)^2/(x^8 - x^7 - x^5 + x^4 - x^3 - x + 1), {x, 0, 50}], x] (* G. C. Greubel, Aug 07 2017 *)
LinearRecurrence[{1,0,1,-1,1,0,1,-1},{1,3,5,8,12,17,25,37,53},50] (* Harvey P. Dale, Jul 26 2024 *)
  • PARI
    Vec((x^2+1)*(x^4+1)*(x+1)^2/(x^8-x^7-x^5+x^4-x^3-x+1) + O(x^100)) \\ Altug Alkan, Dec 29 2015

Formula

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

A265064 Coordination sequence for (2,5,5) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 5, 8, 13, 19, 26, 37, 53, 74, 103, 145, 204, 285, 399, 560, 785, 1099, 1540, 2159, 3025, 4238, 5939, 8323, 11662, 16341, 22899, 32088, 44963, 63005, 88288, 123715, 173357, 242920, 340397, 476987, 668386, 936589, 1312413, 1839042, 2576991, 3611057, 5060060, 7090501, 9935695, 13922576, 19509265, 27337715
Offset: 0

Views

Author

N. J. A. Sloane, Dec 29 2015

Keywords

Links

Crossrefs

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

Programs

  • Mathematica
    CoefficientList[Series[(x^4 + x^3 + x^2 + x + 1) (x + 1)^2 / (x^6 - x^4 - x^3 - x^2 + 1), {x, 0, 45}], x] (* Vincenzo Librandi, Jan 20 2016 *)
LinearRecurrence[{0,1,1,1,0,-1},{1,3,5,8,13,19,26},50] (* Harvey P. Dale, Mar 29 2025 *)
  • PARI
    Vec((x^4+x^3+x^2+x+1)*(x+1)^2/(x^6-x^4-x^3-x^2+1) + O(x^50)) \\ Michel Marcus, Jan 20 2016

Formula

G.f.: (x^4+x^3+x^2+x+1)*(x+1)^2/(x^6-x^4-x^3-x^2+1).
Previous Showing 11-20 of 49 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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