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

A374301 Number of growing self-avoiding walks of length n on a half-infinite strip of height 5 with a trapped endpoint.

Original entry on oeis.org

2, 3, 8, 13, 32, 69, 161, 361, 845, 1846, 4241, 9132, 20791, 44908, 101361, 220149, 493710, 1076528, 2401244, 5248819, 11659368, 25531485, 56546077, 123976603, 274020536, 601294678, 1327099050, 2913847433, 6424359845, 14111695015, 31089757238, 68312316581
Offset: 5

Views

Author

Jay Pantone, Jul 22 2024

Keywords

Comments

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and for which all neighbors of the endpoint are part of the walk, i.e., the endpoint is trapped. This sequence is about GSAWs on the grid graph of integer points (x,y) where x >= 0 and y is in {0,1,2,3,4}. The GSAW must start at the point (0,0). The length of a GSAW is the number of edges.

Examples

			The a(5) = 2 walks are:
>   *  *  *        *  *  *
>
>   *  *  *        *  *  *
>
>   *--*  *        *  *  *
>   |  |
>   *  *  *        *--*--*
>      |           |     |
>   *--*  *        *  *--*

Links

Crossrefs

Cf. A078528, A374299.

Formula

See Links section for generating function.

A374303 Number of growing self-avoiding walks of length n on a half-infinite strip of height 6 with a trapped endpoint.

Original entry on oeis.org

2, 2, 9, 10, 40, 58, 206, 342, 1121, 2024, 6020, 11469, 31574, 62660, 164376, 336835, 853656, 1795319, 4434739, 9511931, 23042967, 50154356, 119696075, 263380585, 621470158, 1378659503, 3225317853, 7199055796, 16732951708, 37523280788, 86787492382
Offset: 5

Views

Author

Jay Pantone, Jul 22 2024

Keywords

Comments

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and for which all neighbors of the endpoint are part of the walk, i.e., the endpoint is trapped. This sequence is about GSAWs on the grid graph of integer points (x,y) where x >= 0 and y is in {0,1,2,3,4,5}. The GSAW must start at the point (0,0). The length of a GSAW is the number of edges.

Examples

			The a(5) = 2 walks are:
>   *  *  *        *  *  *
>
>   *  *  *        *  *  *
>
>   *  *  *        *  *  *
>
>   *--*  *        *  *  *
>   |  |
>   *  *  *        *--*--*
>      |           |     |
>   *--*  *        *  *--*

Links

Crossrefs

Cf. A078528, A374301.

Formula

See Links section for generating function.

A374297 Number of growing self-avoiding walks of length n on a half-infinite strip of height 3 with a trapped endpoint.

Original entry on oeis.org

1, 2, 2, 6, 10, 20, 41, 79, 146, 285, 538, 1039, 1982, 3812, 7272, 13961, 26686, 51161, 97865, 187518, 358835, 687327, 1315616, 2519472, 4823116, 9235610, 17681264, 33855310, 64817361, 124105590, 237610012, 454943624, 871035486, 1667726103, 3193049603
Offset: 4

Views

Author

Jay Pantone, Jul 03 2024

Keywords

Comments

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and for which all neighbors of the endpoint are part of the walk, i.e., the endpoint is trapped. This sequence is about GSAWs on the grid graph of integer points (x,y) where x >= 0 and y is in {0,1,2}. The GSAW must start at the point (0,0). The length of a GSAW is the number of edges.

Examples

			The a(4) = 1 and a(5) = 2 walks are:
  *--*  *    *--*  *    *  *  *
     |       |  |
  *--*  *    *  *  *    *--*--*
  |             |       |     |
  *     *    *--*  *    *  *--*
The GSAW below has length 10.
  *--*--*  *  *  *
        |
  *--*  *--*  *  *
  |  |     |
  *  *--*--*  *  *

Links

Crossrefs

Cf. A078528.

Formula

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

A374299 Number of growing self-avoiding walks of length n on a half-infinite strip of height 4 with a trapped endpoint.

Original entry on oeis.org

3, 2, 9, 8, 36, 45, 153, 235, 658, 1095, 2760, 4994, 11757, 22415, 50587, 99631, 218605, 439382, 947346, 1929565, 4113065, 8450088, 17879748, 36937722, 77783590
Offset: 5

Views

Author

Jay Pantone, Jul 15 2024

Keywords

Comments

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and for which all neighbors of the endpoint are part of the walk, i.e., the endpoint is trapped. This sequence is about GSAWs on the grid graph of integer points (x,y) where x >= 0 and y is in {0,1,2,3}. The GSAW must start at the point (0,0). The length of a GSAW is the number of edges.

Examples

			The a(5) = 3 walks are:
  *--*  *        *  *  *        *  *  *
     |
  *--*  *        *--*  *        *  *  *
  |              |  |
  *  *  *        *  *  *        *--*--*
  |                 |           |     |
  *  *  *        *--*  *        *  *--*

Links

Crossrefs

Cf. A078528.

Formula

G.f.: ((12*x^39 + 14*x^38 - 20*x^37 - 18*x^36 - 45*x^35 - 12*x^34 + 107*x^33 - 38*x^32 + 3*x^31 - 49*x^30 - 38*x^29 + 242*x^28 - 11*x^27 - 66*x^26 - 181*x^25 - 144*x^24 + 246*x^23 + 91*x^22 + 72*x^21 - 208*x^20 - 150*x^19 + 98*x^18 + 57*x^17 + 143*x^16 - 74*x^15 + 5*x^14 - 21*x^13 + 28*x^12 - 17*x^11 - 55*x^10 - 17*x^9 + 22*x^8 + 45*x^7 + 10*x^6 - 19*x^5 - 21*x^4 + 3*x^3 + 7*x^2 + 4*x - 3)*x^5)/((2*x^19 + 2*x^18 - 7*x^17 - 6*x^16 + 5*x^15 + 8*x^14 + 7*x^13 - 17*x^12 - 8*x^11 + 3*x^10 + 10*x^9 + 3*x^8 - 8*x^7 + 2*x^6 - x^5 + 6*x^4 - 3*x^3 - 2*x + 1)*(4*x^20 - 2*x^18 - 5*x^16 + 8*x^14 - x^12 + 2*x^10 - 4*x^8 + 2*x^6 + 3*x^4 - 4*x^2 + 1)).

A374305 Number of growing self-avoiding walks of length n on a half-infinite strip of height 7 with a trapped endpoint.

Original entry on oeis.org

2, 2, 8, 11, 34, 70, 180, 423, 1035, 2557, 6106, 15039, 35538, 85561, 201870, 478444, 1129498, 2654505, 6270807, 14679261, 34662653, 81011176, 191059001, 446245461, 1050699473, 2453328994, 5766594972, 13462400943, 31595520207, 73752506984, 172876421034
Offset: 5

Views

Author

Jay Pantone, Jul 22 2024

Keywords

Comments

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and for which all neighbors of the endpoint are part of the walk, i.e., the endpoint is trapped. This sequence is about GSAWs on the grid graph of integer points (x,y) where x >= 0 and y is in {0,1,2,3,4,5,6}. The GSAW must start at the point (0,0). The length of a GSAW is the number of edges.

Examples

			The a(5) = 2 walks are:
>   *  *  *        *  *  *
>
>   *  *  *        *  *  *
>
>   *  *  *        *  *  *
>
>   *  *  *        *  *  *
>
>   *--*  *        *  *  *
>   |  |
>   *  *  *        *--*--*
>      |           |     |
>   *--*  *        *  *--*

Links

Crossrefs

Cf. A078528, A374303.

Formula

See Links section for generating function.

A078527 Number of maximally 2-constrained walks on square lattice trapped after n steps.

Original entry on oeis.org

0, 1, 9, 7, 3, 36, 26, 13, 1, 100, 54, 19, 7, 247, 147, 68, 27, 12, 552, 294, 151
Offset: 7

Views

Author

Hugo Pfoertner, Nov 27 2002

Keywords

Comments

In a 2D self-avoiding walk there may be steps, where the number of free target positions is less than 3. A step is called k-constrained, if only k<3 neighbors were not visited before. Self-trapping occurs at step n (the next step would have k=0). A maximally 2-constrained n-step walk contains n-floor((4*n+1)^(1/2))-2 steps with k=2 (conjectured). The first step is chosen fixed (0,0)->(1,0), all other steps have k=3. This sequence counts those walks among all possible self-trapping n-step walks A077482(n).

Examples

			a(7)=0 because the unique shortest possible self-trapping walk has no constrained steps. Of the A077482(10)=25 self-trapping walks of length n=10, there are A078528(10)=5 unconstrained walks (9 steps with free choice of direction). a(10)=7 walks are maximally 2-constrained containing 2 steps with k=2. Among the remaining 13 walks there are 11 walks having 1 step with k=2 and 2 walks have 1 forced step k=1. An illustration of all unconstrained and all maximally 2-constrained 10-step walks is given in the first link under "5 Unconstrained and 7 maximally 2-constrained walks of length 10". a(15)=1 is a unique ("perfectly constrained") walk visiting all lattice points of a 4*4 square, see "Examples for walks with the maximum number of constrained steps" provided at the given link.

Links

Crossrefs

Cf. A077482, A076874, A078528, A001411.

Programs

  • Fortran
    c Program provided at given link

Extensions

a(24)-a(27) from Sean A. Irvine, Jul 04 2025

A374304 Number of growing self-avoiding walks with displacement n on a half-infinite strip of height 6 with a trapped endpoint.

Original entry on oeis.org

23, 629, 15134, 323031, 6428665, 122523673, 2267420832, 41081096139, 732520397439, 12900298930153, 224940605616826, 3890634712091201, 66843522591221500, 1141958198925483582, 19416047904038468727, 328765736871514344297, 5547125910154291613320
Offset: 1

Views

Author

Jay Pantone, Jul 22 2024

Keywords

Comments

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and for which all neighbors of the endpoint are part of the walk, i.e., the endpoint is trapped. This sequence is about GSAWs on the grid graph of integer points (x,y) where x >= 0 and y is in {0,1,2,3,4,5}. The GSAW must start at the point (0,0). The displacement of a GSAW is the difference between the largest and smallext x-values that it reaches.

Examples

			Five of the a(1) = 23 walks are:
 *--*  *      *  *  *      *  *  *      *  *  *      *--*  *
 |  |                                                |  |
 *  *  *      *--*  *      *--*  *      *  *  *      *  *  *
 |  |         |  |         |  |                      |  |
 *  *  *      *  *  *      *  *  *      *  *  *      *  *  *
    |         |  |            |                      |  |
 *--*  *      *  *  *      *--*  *      *--*  *      *  *  *
 |               |         |            |  |         |  |
 *  *  *      *--*  *      *--*  *      *  *  *      *  *  *
 |            |               |            |            |
 *  *  *      *  *  *      *--*  *      *--*  *      *--*  *

Links

Crossrefs

Cf. A078528, A374303.

Formula

See Links section for generating function.

A374306 Number of growing self-avoiding walks with displacement n on a half-infinite strip of height 7 with a trapped endpoint.

Original entry on oeis.org

47, 2221, 94006, 3527224, 123159829, 4110628551, 133093672039, 4216993511767, 131454310596858, 4046054885054361, 123275425298494683, 3724935782123793466, 111781579014020685006, 3335061533295212856274, 99013139230297294579692, 2927094675162133314593603
Offset: 1

Views

Author

Jay Pantone, Jul 23 2024

Keywords

Comments

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and for which all neighbors of the endpoint are part of the walk, i.e., the endpoint is trapped. This sequence is about GSAWs on the grid graph of integer points (x,y) where x >= 0 and y is in {0,1,2,3,4,5,6}. The GSAW must start at the point (0,0). The displacement of a GSAW is the difference between the largest and smallext x-values that it reaches.

Examples

			Five of the a(1) = 47 walks are:
 *--*  *      *  *  *      *  *  *      *  *  *      *--*  *
 |  |                                                |  |
 *  *  *      *  *  *      *  *  *      *  *  *      *  *  *
 |  |                                                |  |
 *  *  *      *--*  *      *--*  *      *  *  *      *  *  *
 |  |         |  |         |  |                      |  |
 *  *  *      *  *  *      *  *  *      *  *  *      *  *  *
    |         |  |            |                      |  |
 *--*  *      *  *  *      *--*  *      *--*  *      *  *  *
 |               |         |            |  |         |  |
 *  *  *      *--*  *      *--*  *      *  *  *      *  *  *
 |            |               |            |            |
 *  *  *      *  *  *      *--*  *      *--*  *      *--*  *

Links

Crossrefs

Cf. A078528, A374305.

Formula

See Links section for generating function.

A374298 Number of growing self-avoiding walks with displacement n on a half-infinite strip of height 3 with a trapped endpoint.

Original entry on oeis.org

2, 10, 40, 148, 526, 1828, 6256, 21190, 71260, 238432, 794914, 2643352, 8773684, 29082010, 96303640, 318678388, 1053993646, 3484654468, 11517602176, 38060746390, 125756057260, 415464635392, 1372477613794, 4533688494712, 14975452784164, 49464657237610
Offset: 1

Views

Author

Jay Pantone, Jul 03 2024

Keywords

Comments

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and for which all neighbors of the endpoint are part of the walk, i.e., the endpoint is trapped. This sequence is about GSAWs on the grid graph of integer points (x,y) where x >= 0 and y is in {0,1,2}. The GSAW must start at the point (0,0). The displacement of a GSAW is the difference between the largest and smallext x-values that it reaches.

Examples

			The a(1) = 2 walks are:
  *--*  *      *--*  *
     |         |  |
  *--*  *      *  *  *
  |               |
  *  *  *      *--*  *

Links

Crossrefs

Cf. A078528, A374297.

Formula

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

A374300 Number of growing self-avoiding walks with displacement n on a half-infinite strip of height 4 with a trapped endpoint.

Original entry on oeis.org

5, 44, 330, 2231, 14234, 87670, 526549, 3105097, 18061476, 103955447, 593388315, 3364743202, 18977238539, 106562551704, 596209056866, 3325672377580, 18503794814297, 102734584002260, 569364274759972, 3150649232873918, 17411856639412771, 96118767225465184
Offset: 1

Views

Author

Jay Pantone, Jul 16 2024

Keywords

Comments

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and for which all neighbors of the endpoint are part of the walk, i.e., the endpoint is trapped. This sequence is about GSAWs on the grid graph of integer points (x,y) where x >= 0 and y is in {0,1,2,3}. The GSAW must start at the point (0,0). The displacement of a GSAW is the difference between the largest and smallext x-values that it reaches.

Examples

			The a(1) = 5 walks are:
 *--*  *      *--*  *      *--*  *      *  *  *      *--*  *
    |         |  |            |                      |  |
 *--*  *      *  *  *      *--*  *      *--*  *      *  *  *
 |               |         |            |  |         |  |
 *  *  *      *--*  *      *--*  *      *  *  *      *  *  *
 |            |               |            |            |
 *  *  *      *  *  *      *--*  *      *--*  *      *--*  *

Links

Crossrefs

Cf. A078528, A374299.

Formula

G.f.: (-(11*x^12+4*x^11-138*x^10+205*x^9+119*x^8-552*x^7+485*x^6-93*x^5-112*x^4+132*x^3-85*x^2+31*x-5)*x)/((x^6+2*x^5-9*x^4-5*x^3+15*x^2-8*x+1)*(2*x^5+3*x^4-7*x^3+12*x^2-7*x+1)).
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