A259808 -id:A259808 - cp's OEIS frontend

A337400 Table read by antidiagonals: T(w,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a tube of cross section 2w X 2w where the walk starts at the middle of the tube.

6, 26, 6, 98, 30, 6, 330, 146, 30, 6, 1130, 658, 150, 30, 6, 3746, 2858, 722, 150, 30, 6, 12802, 11802, 3450, 726, 150, 30, 6, 42498, 48282, 15930, 3530, 726, 150, 30, 6, 143610, 193714, 72522, 16826, 3534, 726, 150, 30, 6, 472242, 781114, 321794, 80010, 16922, 3534, 726, 150, 30, 6

Offset: 1

Scott R. Shannon, Aug 26 2020

			T(1,2) = 26 as after a step in one of the four directions toward the tube's side the walk must turn along the side; this eliminates the 2-step straight walk in those four directions, so the total number of walks is 6*5 - 4 = 26.
The table begins:
6 26  98 330 1130  3746 12802  42498  143610  472242  1570714   5110426  16779354...
6 30 146 658 2858 11802 48282 193714  781114 3114890 12508114  49767002 199252346...
6 30 150 722 3450 15930 72522 321794 1415450 6134650 26527690 113725546 487875250...
6 30 150 726 3530 16826 80010 373962 1736538 7946946 36158802 162796866 730521658...
6 30 150 726 3534 16922 81274 386138 1833018 8615906 40370370 187477426 867587114...
6 30 150 726 3534 16926 81386 387834 1851546 8780162 41630146 196172338 923017178...
6 30 150 726 3534 16926 81390 387962 1853738 8806962 41893346 198386594 939630954...
6 30 150 726 3534 16926 81390 387966 1853882 8809714 41930594 198788354 943314378...
6 30 150 726 3534 16926 81390 387966 1853886 8809874 41933970 198838482 943903786...
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934146 198842546 943969482...
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842738 943974298...
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842742 943974506...
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842742 943974510...

Cf. A337401 (start at center of tube's side), A337403 (start at tube's edge), A001412 (w->infinity), A116904, A337023, A259808, A039648.

For n <= w, T(w,n) = A001412(n).

A335806 The number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size 2x2x2 where the walk starts at the middle of the box's edge.

Original entry on oeis.org

1, 4, 12, 40, 118, 358, 936, 2600, 6212, 16068, 34936, 83708, 163452, 357056, 613592, 1205716, 1770616, 3073480, 3715920, 5573480, 5255048, 6591160, 4353912, 4330096, 1513712, 1061392, 0

Offset: 0

Scott R. Shannon, Aug 14 2020

			a(1) = 4 as the walk is free to move one step in four directions.
a(2) = 12. A first step along either edge leading to the corner leaves two possible second steps. A first step to the centre of either face can be followed by a second step to three edges or to the center of the cube, four steps in all. Thus the total number of 2-step walks is 2*2+2*4 = 12.
a(26) = 0 as it is not possible to visit all 26 available lattice points when the walk starts from the middle of the box's edge.

Cf. A336862 (other box sizes), A337021 (start at center of box), A337033 (start at center of face), A337034 (start at corner of box), A001412, A259808, A039648.

For n>=27 all terms are 0 as the walk contains more steps than there are available lattice points in the 2x2x2 box.

A336862 Table read by antidiagonals: T(h,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size 2h X 2h X 2h where the walk starts at the middle of the box's edge.

Original entry on oeis.org

4, 12, 4, 40, 14, 4, 118, 54, 14, 4, 358, 208, 56, 14, 4, 936, 826, 224, 56, 14, 4, 2600, 3232, 936, 226, 56, 14, 4, 6212, 12688, 3862, 956, 226, 56, 14, 4, 16068, 48924, 16196, 4026, 958, 226, 56, 14, 4, 34936, 187276, 67346, 17246, 4050, 958, 226, 56, 14, 4

Offset: 1

Scott R. Shannon, Aug 14 2020

			T(1,2) = 12. A first step along either edge leading to the corner leaves two possible second steps. A first step to the center of either face can be followed by a second step to three edges or to the center of the box, four steps in all. Thus the total number of 2-step walks is 2*2+2*4 = 12.
.
The table begins:
.
4 12 40 118 358  936  2600  6212  16068   34936   83708   163452    357056...
4 14 54 208 826 3232 12688 48924 187276  705196 2627950  9670620  35231628...
4 14 56 224 936 3862 16196 67346 282676 1180326 4950936 20646098  86165926...
4 14 56 226 956 4026 17246 73588 316456 1358518 5860464 25266192 109288486...
4 14 56 226 958 4050 17478 75288 327778 1425340 6236152 27260378 119641050...
4 14 56 226 958 4052 17506 75600 330362 1444544 6360718 28020896 123963354...
4 14 56 226 958 4052 17508 75632 330766 1448280 6391426 28238732 125405300...
4 14 56 226 958 4052 17508 75634 330802 1448788 6396618 28285548 125766436...
4 14 56 226 958 4052 17508 75634 330804 1448828 6397242 28292536 125835068...
4 14 56 226 958 4052 17508 75634 330804 1448830 6397286 28293288 125844228...
4 14 56 226 958 4052 17508 75634 330804 1448830 6397288 28293336 125845120...
4 14 56 226 958 4052 17508 75634 330804 1448830 6397288 28293338 125845172...
4 14 56 226 958 4052 17508 75634 330804 1448830 6397288 28293338 125845174...

Cf. A259808 (h->infinity), A335806 (h=1), A337023 (start at center of box), A337031 (start at center of face), A337035 (start at corner of box), A001412, A039648.

For n <= h, T(h,n) = A259808(n).
Row 1 = T(1,n) = A335806(n).
For n >= (2h+1)^3, T(h,n) = 0 as the walk contains more steps than there are available lattice points in the 2h X 2h X 2h box.

A337401 Triangle read by rows: T(n,w) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a tube of cross section 2w X 2w where the walk starts at the center of the tube's side.

Original entry on oeis.org

5, 19, 21, 72, 91, 93, 258, 383, 407, 409, 926, 1638, 1821, 1851, 1853, 3176, 6856, 8019, 8295, 8331, 8333, 11000, 28810, 35506, 37531, 37921, 37963, 37965, 36988, 119106, 155492, 168399, 171691, 172215, 172263, 172265, 125302, 492766, 683126, 758182, 781811, 786823, 787501, 787555, 787557

Offset: 1

Scott R. Shannon, Aug 26 2020

			T(2,1) = 19 as after a step in one of the two directions toward the adjacent tube side the walk must turn along the side; this eliminates the 2-step straight walk in those two directions, so the total number of walks is 4*4 + 5 - 2 = 19.
The table begins:
5;
19,21;
72,91,93;
258,383,407,409;
926,1638,1821,1851,1853;
3176,6856,8019,8295,8331,8333;
11000,28810,35506,37531,37921,37963,37965;
36988,119106,155492,168399,171691,172215,172263,172265;
125302,492766,683126,758182,781811, 786823,787501,787555,787557;
414518,2013142,2981996,3393526,3545117,3585297,3592551,3593403,3593463,3593465;

Scott R. Shannon, Data for n=1 to n=14.

Cf. A337400 (start at middle of tube), A337403 (start at tube's edge), A116904 (w->infinity), A001412, A337023, A259808, A039648.

For w>=n, T(n,w) = A116904(n).

A337403 Table read by antidiagonals: T(w,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a tube of cross section w x w where the walk starts at the tube's edge.

Original entry on oeis.org

4, 12, 4, 36, 14, 4, 98, 54, 14, 4, 274, 200, 56, 14, 4, 702, 744, 224, 56, 14, 4, 1854, 2626, 926, 226, 56, 14, 4, 4614, 9186, 3738, 956, 226, 56, 14, 4, 11778, 31122, 15056, 4014, 958, 226, 56, 14, 4, 28914, 105766, 59092, 17074, 4050, 958, 226, 56, 14, 4

Offset: 1

Scott R. Shannon, Aug 26 2020

			T(1,2) = 12 as after a step in one of the two directions toward the adjacent tube edge the walk must turn along the side; this eliminates the 2-step straight walk in those two directions, so the total number of walks is 2*3 + 2*4 - 2 = 12.
The table begins:
4 12 36  98 274  702  1854  4614  11778   28914   72394   176310    435346...
4 14 54 200 744 2626  9186 31122 105766  351798 1175726  3859350  12729142...
4 14 56 224 926 3738 15056 59092 230254  881850 3367124 12712194  47952018...
4 14 56 226 956 4014 17074 71774 301578 1251362 5170636 21143094  86148002...
4 14 56 226 958 4050 17464 75060 325064 1400650 6040372 25882446 110668184...
4 14 56 226 958 4052 17506 75584 330070 1440668 6321926 27685144 121407404...
4 14 56 226 958 4052 17508 75632 330748 1447916 6386092 28180426 124857572...
4 14 56 226 958 4052 17508 75634 330802 1448768 6396174 28278426 125681952...
4 14 56 226 958 4052 17508 75634 330804 1448828 6397220 28292004 125825794...
4 14 56 226 958 4052 17508 75634 330804 1448830 6397286 28293264 125843600...
4 14 56 226 958 4052 17508 75634 330804 1448830 6397288 28293336 125845094...
4 14 56 226 958 4052 17508 75634 330804 1448830 6397288 28293338 125845172...
4 14 56 226 958 4052 17508 75634 330804 1448830 6397288 28293338 125845174...

Cf. A337400 (start at middle of tube), A337401 (start at center of tube's side) A259808 (w->infinity), A001412, A337023, A259808, A039648.

For n <= w, T(w,n) = A259808(n).

cp's OEIS Frontend

A337400 Table read by antidiagonals: T(w,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a tube of cross section 2w X 2w where the walk starts at the middle of the tube.

Views

Author

Keywords

Examples

Crossrefs

Formula

A335806 The number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size 2x2x2 where the walk starts at the middle of the box's edge.

Views

Author

Keywords

Examples

Crossrefs

Formula

A336862 Table read by antidiagonals: T(h,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size 2h X 2h X 2h where the walk starts at the middle of the box's edge.

Views

Author

Keywords

Examples

Crossrefs

Formula

A337401 Triangle read by rows: T(n,w) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a tube of cross section 2w X 2w where the walk starts at the center of the tube's side.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A337403 Table read by antidiagonals: T(w,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a tube of cross section w x w where the walk starts at the tube's edge.

Views

Author

Keywords

Examples

Crossrefs

Formula