A359073
Sum of square end-to-end displacements over all n-step self-avoiding walks of A359709.
Original entry on oeis.org
0, 4, 16, 44, 160, 556, 1744, 12252, 15840, 98876, 138160, 709900, 1155616, 5098260, 11820656, 37085908, 111147104, 281078764, 932893104, 2255139900, 7295211968, 18928121236, 54864568720, 160016686500, 404167501888, 1331607134172, 2945597090384, 10805511468852, 21448743511648
Offset: 0
A359709
Number of n-step self-avoiding walks on a 2D square lattice whose end-to-end distance is an integer.
Original entry on oeis.org
1, 4, 4, 12, 28, 76, 164, 732, 1044, 4924, 6724, 30636, 43972, 190516, 313996, 1197908, 2284260, 7678188, 16257604, 50524252, 113052396, 341811828, 773714436, 2358452388, 5245994292, 16447462492, 35395532236, 115129727188, 238542983748, 804980005276
Offset: 0
a(3) = 12 as, in the first quadrant, there is one 3-step SAW whose end-to-end distance is an integer (1 unit):
.
X---.
|
X---.
.
This can be walked in 8 different ways on a 2D square lattice. There are also the four walks directly along the x and y axes, giving a total of 8 + 4 = 12 walks.
A336726
Number of n-step self-avoiding walks on the half-Manhattan lattice with no non-contiguous adjacencies.
Original entry on oeis.org
1, 3, 7, 13, 25, 47, 89, 169, 321, 599, 1125, 2097, 3925, 7317, 13667, 25389, 47313, 87781, 163265, 302617, 562023, 1040465, 1929879, 3569627, 6613725, 12223777, 22626357, 41790173, 77289525, 142665435, 263661315, 486420171, 898372311, 1656580751, 3057774587, 5636030345
Offset: 0
A337441
Number of n-step self-avoiding walks on a 2D square lattice where the walk consists of three different units and each unit cannot be adjacent to another unit of the same type.
Original entry on oeis.org
1, 4, 12, 28, 68, 164, 396, 956, 2292, 5420, 12924, 30812, 73228, 174228, 413092, 971900, 2299244, 5440924, 12846900, 30355228, 71572196, 167933164, 395458372, 931516756, 2191050916, 5156589252, 12118552572, 28383666716, 66646232884, 156526277324, 367254003324, 862071250300, 2021536511948
Offset: 0
The walk consists of three different units:
.
... --A--B--C--A--B--C--A--B--C-- ...
.
The one forbidden 4-step walk in the first quadrant is:
.
A---C
|
A---B
.
as two A units cannot be adjacent. As this walk can be taken in eight different ways on the square lattice a(3) = 4*8 + 4 - 8 = A001411(3) - 8 = 28;
The two forbidden 4-step walks are:
.
C---A B---A
| | |
A---B B A---B---C
.
as two B unit cannot be adjacent. These, along with the forbidden 3-step walk, remove four 4-step walks so a(4) = 12*8 + 4 - 8*4 = A001411(4) - 32 = 68.
Three forbidden 5-step walks are:
.
B---A
| | A---B C---B
C C | | |
| A---B---C C A---B---C---A
A---B
.
as two C units cannot be adjacent.
Up to n=6 this sequence matches A173380 as the later excludes the above same walks as it does not allow any adjacencies. However for n=7 the below two first-quadrant walks are allowed in this sequence:
.
A---C---B C---B---A
| | | |
B A A C
| | |
A---B---C B A---B
.
as the A and B units, being different, can be adjacent. These same walks are forbidden in A173380. As each of these can be taken in 8 ways on the square lattice a(7) = A173380(7) + 2*8 = 940 + 16 = 956.
A336758
Number of n-step self-avoiding walks on the honeycomb lattice with no non-contiguous adjacencies.
Original entry on oeis.org
1, 3, 6, 12, 24, 42, 78, 144, 264, 486, 894, 1620, 2964, 5376, 9798, 17760, 32292, 58398, 105960, 191466, 346854, 626172, 1132800, 2043246, 3692406, 6655068, 12015126, 21641526, 39039810, 70277016, 126682584, 227928780, 410605008, 738423492, 1329477732
Offset: 0
A337367
Sum of square end-to-end distance over all self-avoiding n-step walks on a square lattice where no adjacent points are allowed, except those for consecutive steps.
Original entry on oeis.org
0, 4, 32, 156, 608, 2116, 6816, 20844, 61376, 175628, 491248, 1349172, 3650144, 9751532, 25774672, 67501556, 175375136, 452454276, 1160098576, 2958123556, 7505767840, 18959922796, 47701159264, 119570463980, 298719578688, 743984084700, 1847709517360, 4576818079076, 11309417827072
Offset: 0
The allowed 4-step walks with their associated end-to-end square distances are:
.
+ 10
4 | 8 8 8 16
+--+ + +--+ + + X--+---+---+---+
| | | 10 | |
+ + + +--+--+ +--+ + +--+ 10 + 10
| | | | | | | |
X--+ X--+ X--+ X--+ X--+ X--+--+ X--+--+ X--+--+--+
.
The eight non-straight walks sum to 68, and these can be walked in eight ways on the square lattice. The remaining straight walk can be walking in four ways. Thus a(4) = 68 * 8 + 16 * 4 = 608.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes the sequence A173380).
A337456
Number of n-step self-avoiding walks on a 3D cubic lattice where the walk consists of three different units and each unit cannot be adjacent to another unit of the same type.
Original entry on oeis.org
1, 6, 30, 126, 534, 2262, 9534, 40254, 169302, 702510, 2929806, 12222414, 50908158, 212134902, 882794118, 3654001326, 15159263934, 62906444238, 260853828438, 1081924309806, 4484440327350
Offset: 1
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