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 1-4 of 4 results.

A118313 Sum of squared end-to-end distances of all n-step self-avoiding walks on the simple cubic lattice.

Original entry on oeis.org

0, 6, 72, 582, 4032, 25566, 153528, 886926, 4983456, 27401502, 148157880, 790096950, 4166321184, 21760624254, 112743796632, 580052260230, 2966294589312, 15087996161382, 76384144381272, 385066579325550, 1933885653380544, 9679153967272734, 48295148145655224, 240292643254616694, 1192504522283625600, 5904015201226909614, 29166829902019914840, 143797743705453990030, 707626784073985438752, 3476154136334368955958, 17048697241184582716248, 83487969681726067169454, 408264709609407519880320, 1993794711631386183977574, 9724709261537887936102872, 47376158929939177384568598, 230547785968352575619933376
Offset: 0

Views

Author

R. J. Mathar, May 14 2006

Keywords

Comments

Number of walks is A001412(n).
a(5) is 25556 according to MacDonald et al., but 25566 according to Clisby et al. and is therefore conjectural for now. - R. J. Mathar, Aug 31 2007
Confirmed that a(5) is 25566 [from Nathan Clisby]. Right-hand column, table, p.5 of Schram.

Crossrefs

Extensions

a(5) corrected by Nathan Clisby, Nov 24 2010
a(14), a(22) corrected by Hugo Pfoertner, Aug 13 2011

A078605 Sum of square displacements over all self-avoiding n-step walks on the cubic lattice with the first step specified. Numerator of mean square displacement s(n)=a(n)/(A001412(n)/6).

Original entry on oeis.org

1, 12, 97, 672, 4261, 25588, 147821, 830576, 4566917, 24692980, 131682825, 694386864, 3626770709, 18790632772, 96675376705, 494382431552, 2514666026897, 12730690730212, 64177763220925, 322314275563424, 1613192327878789, 8049191357609204, 40048773875769449, 198750753713937600
Offset: 1

Views

Author

Hugo Pfoertner, Dec 09 2002

Keywords

Comments

A comparison with the conjectured asymptotic behavior of the mean square displacement s(n) over all n-step self-avoiding walks given in Weisstein's article is shown in "Asymptotic Behavior of Mean Square Displacement" at the Pfoertner link.

Examples

			a(2)=12 because the A001412(2)/6 = 5 different self-avoiding 2-step walks end at (1,0,-1), (1,0,1), (1,-1,0), (1,1,0)->d^2=2 and at (2,0,0)->d^2=4. a(2) = 4*2 + 1*4 = 12. See also "Distribution of end point distance" at first link.
		

References

Crossrefs

Cf. A001412, A078717, A079156 (corresponding Manhattan distance sum).
Equals A118313/6.

Programs

  • Fortran
    c Program for distance counting available at Pfoertner link.

Formula

a(n) = Sum_{L=1..A001412(n)/6} ( i_L^2 + j_L^2 + k_L^2 ) where (i_L, j_L, k_L) are the endpoints of all different self-avoiding n-step walks.

Extensions

Terms a(19)-a(36) taken from A118313 by Hugo Pfoertner, Aug 20 2014
Name amended by Scott R. Shannon, Sep 17 2020

A323857 Sum of end-to-end Manhattan distances over all self-avoiding n-step walks on 4-d cubic lattice.

Original entry on oeis.org

1, 14, 135, 1144, 9083, 69690, 522781, 3864524, 28243251, 204687550, 1473038447, 10542725976, 75096139471, 532846305962, 3767808141891, 26566180648012, 186826646453453
Offset: 1

Views

Author

Hugo Pfoertner, Feb 03 2019

Keywords

Comments

The first step is kept fixed, i.e., (0,0,0,0) -> (1,0,0,0).

Examples

			a(3) = 135, because there are 6 (of A010575(3)/8=49) end points with Manhattan distance 1, (0,-1,0,0), (0,1,0,0), (0,0,-1,0), (0,0,1,0), (0,0,0,-1), (0,0,0,1), and the remaining 43 end points all have Manhattan distance 3, e.g., (3,0,0,0), (2,-1,0,0), ..., (1,-1,-1,0), ... 135 = 6*1 + 43*3.
		

Crossrefs

A079158 Sum of end-to-end Manhattan distances over all self-avoiding walks on cubic lattice trapped after n steps.

Original entry on oeis.org

5, 40, 399, 2472, 17436, 98400, 601626, 3238694, 18355742, 96020478
Offset: 11

Views

Author

Hugo Pfoertner, Dec 30 2002

Keywords

Comments

Mean Manhattan displacement is a(n)/A077817(n).
See also "Comparison of average Euclidean and Manhattan displacements" at link

Examples

			a(12)=40 because the A077817(12)=20 trapped walks stop at 5*(1,1,0)->d=2, 5*(2,0,0)->d=2, 10*(1,0,1)->d=2, so a(12)=5*2+5*2+10*2=40. See "Enumeration of all self-trapping walks of length 12" at link.
		

Crossrefs

Cf. A077817, A079156, A079157 (corresponding squared distance sum).

Programs

  • Fortran
    c Program for distance counting available at link.

Formula

a(n)= Sum_{l=1..A077817(n)} (|i_l| + |j_l| + |k_l|) where (i_l, j_l, k_l) are the end points of all different self-avoiding walks trapped after n steps.

Extensions

a(19)-a(20) from Sean A. Irvine, Jul 31 2025
Showing 1-4 of 4 results.