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 1-5 of 5 results.

A078528 Number of unconstrained walks on square lattice trapped after n steps.

Original entry on oeis.org

1, 1, 2, 5, 15, 30, 76, 170, 422, 961, 2339, 5390, 12977, 30059, 71918, 167019, 397691, 924931, 2194478, 5107991, 12085695, 28143758, 66442935, 154759821, 364706675, 849562628
Offset: 7

Views

Author

Hugo Pfoertner, Nov 27 2002

Keywords

Comments

See under A078527. In the probability sum in A077483 and A078526 the unconstrained walks are responsible for the occurrence of 3^(n-1) in the denominator of P(n).

Examples

			a(7)=1 because the unique shortest walk contains no constrained steps. a(10)=5: See illustration in "5 Unconstrained and 7 maximally 2-constrained walks of length 10" given at link.

Links

Crossrefs

Cf. A077482, A077483, A078526, A078527, A001411.

Programs

  • Fortran
    c Program provided at given link

Extensions

a(24)-a(32) from Sean A. Irvine, Jul 03 2025

A077484 Probability P(n) of the occurrence of a 2D self-trapping walk of length n: Exponent of 2 in the denominator.

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 2, 4, 5, 2, 5, 7, 8, 8
Offset: 7

Views

Author

Hugo Pfoertner, Nov 08 2002

Keywords

Comments

This sequence is the exponent of 2 in the denominator for P(n).

References

Crossrefs

Cf. A077483.

Formula

P(n) = A077483(n) / ( 3^(n-1) * 2^A077484(n) )

A076874 a(n) = n - floor ( ( 4*n + 1 )^(1/2) ).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39, 40, 41, 41, 42, 43, 44, 45, 46, 47, 48, 48, 49, 50, 51, 52, 53, 54, 55, 55, 56, 57, 58, 59
Offset: 3

Views

Author

Hugo Pfoertner, Nov 24 2002

Keywords

Comments

Conjecture: For n>=7, a(n)-2 is the maximum number of steps in a 2D self-avoiding random walk trapped after n steps having only 2 choices for the next step. a(n) >= A077484(n) + 2.

Links

Crossrefs

Cf. A077483, A077484, A001411.

Programs

A078526 Probability P(n) of the occurrence of a 2D self-trapping walk of length n.

Original entry on oeis.org

1, 5, 31, 173, 1521, 4224, 33418, 184183, 1370009, 3798472, 26957026, 150399317, 1034714947, 2897704261, 19494273755, 109619578524, 724456628891
Offset: 7

Views

Author

Hugo Pfoertner, Nov 27 2002

Keywords

Comments

This is a cleaner representation than the one given by A077483 and A077484, using the upper bound for the denominator A077484 given in A076874.

Examples

			See under A077483; the inclusion of a(7)=1 is somewhat artificial due to the occurrence of 2^(-1) in the denominator: P(7)=a(7)/(3^6 *2^(7-floor(sqrt(29))-3))= 1/(729*2^(7-5-3))=1/(729*2*(-1))=2/729 See also: "Count self-trapping walks up to length 23" provided at given link.

References

Links

Crossrefs

Cf. A077483, A077484, A076874, A001411.

Programs

  • Fortran
    c Program provided at given link

Formula

P(n) = a(n) / ( 3^(n-1) * 2^(n-floor((4*n+1)^(1/2))-3) ) = a(n) / ( 3^(n-1) * 2^(A076874(n)-3) )

A381979 Decimal expansion of the expected number of steps to termination by self-trapping of a self-avoiding random walk on the square lattice.

Original entry on oeis.org

7, 0, 7, 5, 9
Offset: 2

Views

Author

Yi Yang, Mar 11 2025

Keywords

Comments

The average walk length determined by 1.2*10^12 simulations is 70.75915 +- 0.00010

Examples

			70.759...

References

Links

Crossrefs

Cf. A378903 (The expected walk length on the cubic lattice).
Cf. A077483 (Probability of the occurrence of each walk length).
Cf. A322831.
Showing 1-5 of 5 results.

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

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